NON-COMPUTABLE JULIA SETS
arXiv:math/0406416v1 [math.DS] 22 Jun 2004
NON-COMPUTABLE JULIA SETS M. BRAVERMAN, M. YAMPOLSKY Abstract. We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
1. Summary of the paper Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most na¨ıve work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor’s distance estimator algorithm [Mil], uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show: Main Theorem There exists a parameter value c ∈ C such that the Julia set of the quadratic polynomial fc (z) = z 2 + c is not computable. The structure of the paper is as follows. In the Introduction we discuss the question of computability of real sets, and make the relevant definitions. Further in this section we briefly introduce the reader to the main concepts of Complex Dynamics, and discuss the properties of Julia sets relevant to us. In the end of the Introduction, we outline the conceptual idea of the proof of Main Theorem. Section §3 contains the technical lemmas on which the argument is based. In §4 we complete the proof. Acknowledgement. We wish to thank Arnaud Ch´eritat for helpful comments on an earlier draft of this paper. The first author would like to thank Stephen Cook for many discussions on computability of real sets. Date: March 6, 2008. The first author’s research is supported by an NSERC CGS scholarship. The second author’s research is supported by NSERC operating grant. 1
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2. Introduction 2.1. Introduction to the Computability of Real Sets. Classical Computability. The computability theory in general allows us to classify problems into the tractable (“computable”) and intractable (“uncomputable”). All the common computational tasks such as integer operations, list sorting, etc. are easily seen to be computable. On the other hand, there are many uncomputable problems. In the formal setting for the study of computability theory computations are performed by objects called the Turing Machines. Turing Machines were introduced in 1936 by Alan Turing (see [Tur]) and is accepted by the scientific community as the standard model of computation. The Turing Machine (TM in short) is capable of solving exactly the same problems as an ordinary computer. Most of the time, one can think of the TM as a computer program written in any programming language. It is important to mention that there are only countably many TMs, which can be enumerated in a natural way. See [Sip] for a formal discussion on TMs. We define computability as follows. Definition 2.1. We say that a function f : {0, 1}∗ → {0, 1}∗ is computable if there is a TM M, which on input string s outputs the string f (s). We say that the set L ⊂ {0, 1}∗ is computable or decidable if its characteristic function χL : {0, 1}∗ → {0, 1} is computable. While most “common” functions are computable, there are uncountably many uncomputable functions and undecidable sets. The most famous intractable problems are the Halting Problem and the solvability of a Diophantine equation (Hilbert’s 10-th problem), see [Sip] and [Mat] for more information.
Computability of Real Functions and Sets. In the present paper we are interested in the computability of functions f : Rn → R and subsets of Rn , particularly subsets of R2 ∼ = C. We cannot directly apply definition 2.1 here, since real number cannot be represented in general by finite sequences of bits. Denote by D the set of the dyadic rationals, i.e. rationals of the form 2pm . Rationals in D can be easily represented as binary numbers. We say that φ : N → D is an oracle for a real number x, if |x − φ(n)| < 2−n for all n ∈ N. In other words, φ provides a good dyadic approximation for x. We say that a TM M φ is an oracle machine, if at any step of the computation M is allowed to query the value φ(n) for any n. This definition allows us to define the computability of real functions on compact sets. Definition 2.2. We say that a function f : [a, b] → [c, d] is computable, if there exits an oracle TM M φ (m) such that if φ is an oracle for x ∈ [a, b], then on input m, M φ outputs a y ∈ D such that |y − f (x)| < 2−m . In other words, with an access to arbitrarily good approximations for x, M should be able to produce an arbitrarily good approximation for f (x). This definition trivially generalizes to domains of higher dimension. See [Ko1] for more details. One of the most important properties of computable functions is that
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Proposition 2.1. Computable functions are continuous. Let K ⊂ Rk be a compact set. We would like to give a definition for K being computable. Note that saying by analogy to definition 2.1, that C is computable if and only if the characteristic function χC is computable does not work here. Recall that only continuous functions can be computable, hence χK is not computable unless K = ∅. We say that a TM M computes the set K if it approximates K in the Hausdorff metric. Recall that the Hausdorff metric is a metric on compact subsets of Rn defined by (2.1)
dH (X, Y ) = inf{ǫ > 0 | X ⊂ Uǫ (Y ) and Y ⊂ Uǫ (X)},
where Uǫ (S) is defined as the union of the set of ǫ-balls with centers in S. We approximate C using a class C of sets which is dense in metric dH among the compact sets and which has a natural correspondence to binary strings. Namely C is the set of finite unions of dyadic balls: (n ) [ C= B(di , ri ) | where di , ri ∈ D . i=1
Members of C can be encoded as binary strings in a natural way. The following definition is equivalent to the set computability definition given in [Wei] (see also [RW]).
Definition 2.3. We say that a compact set K ⊂ Rk is computable, if exists a TM M(m), such that on input m, M(m) outputs an encoding of Cm ∈ C such that dH (K, Cm ) < 2−m .
To illustrate the robustness of this definition we present the following two equivalent characterizations of computable sets. The first one relates the definition to computer graphics. It is not stated precisely here, but it can be easily made precise. The second one relates the computability of sets to the computability of functions as per definition 2.2. Theorem 2.2. For a compact K ⊂ Rk the following are equivalent: (1) K is computable as per definition 2.3, (2) (in the case k = 1, 2) K can be drawn on a computer screen with arbitrarily good precision, (3) the distance function dK (x) = inf{|x − y| | y ∈ K} is computable as per definition 2.2. In the present paper we are interested in questions concerning the computability of the Julia set Jc = J(fc ) = J(z 2 + c) (see the next section for the definition). Since there are uncounably many possible parameter values for c, we cannot expect for each c to have a machine M such that M computes Jc (recall that there are countably many TMs). On the other hand, it is reasonable to want M to compute Jc with an oracle access to c. Define the function J : C → K ∗ (K ∗ is the set of all compact subsets of C) by J(c) = J(fc ). In a complete analogy to definition 2.2 we can define Definition 2.4. We say that a function f : S → K ∗ for some bounded set S is computable, if there exits an oracle TM M φ (m) such that if φ is an oracle for x ∈ S, then on input m, M φ outputs a C ∈ C such that dH (C, f (x)) < 2−m .
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In the case of Julia sets: Definition 2.5. We say that Jc is computable if the function J : d 7→ Jd is computable on the set {c}. The following has been shown (see [Brv], [Ret]): Theorem 2.3. Denote by H the set of parameters c for which Jc is hyperbolic, then (i) Jc is computable for all c ∈ H, moreover (ii) the function J is computable on B(0, b) ∩ H for all b > 0. Our goal in this paper is to show that there are values of c for which Jc is not computable under definition 2.5, which is the weakest possible definition in this setting. We will be using the following version of theorem 2.1 for set functions. Theorem 2.4. Suppose that a TM M φ computes the function J on a set S, then J is continuous on S in Hausdorff metric. In the next section we proceed to define Julia sets of rational maps and review their basic properties. In particular, towards the end of the introduction, we will see a mechanism by which the continuity required by Theorem 2.4 may fail. It should be noted that the question of computability of dynamically generated fractal sets, such as Julia sets, has been discussed by Blum, Cucker, Shub, and Smale in [BCSS]. The definition of set computability used in [BCSS] is, however, quite different from definition 2.3. The BCSS model allows precise arithmetic, but requires completely accurate pictures to be generated. Under this definition all Julia sets but the most trivial ones can be shown to be uncomputable. Our definition, however, reflects the hardness of generating pictures of the set, and the fact that we have good images of Julia sets in some cases shows the usefulness of this definition in reflecting the true “hardness” of the set. 2.2. Julia sets of polynomial mappings. We recall the main definitions of complex dynamics relevant to our result only briefly; a good general reference is the book of Milnor [Mil]. For a rational mapping R of degree deg R = d ≥ 2 considered as a dynamical system on the Riemann sphere ˆ →C ˆ R:C the Julia set is defined as the complement of the set where the dynamics is Lyapunov-stable: ˆ having an open neighborhood U(z) Definition 2.6. Denote F (R) the set of points z ∈ C n on which the family of iterates R |U (z) is equicontinuous. The set F (R) is called the Fatou ˆ \ F (R) is the Julia set. set of R and its complement J(R) = C In the case when the rational mapping is a polynomial P (z) = a0 + a1 z + · · · + ad z d : C → C
an equivalent way of defining the Julia set is as follows. Obviously, there exists a neighˆ on which the iterates of P uniformly converge to ∞. Denoting A(∞) borhood of ∞ on C
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the maximal such domain of attraction of ∞ we have A(∞) ⊂ F (R). We then have J(P ) = ∂A(∞). ˆ \ cl A(∞) is called the filled Julia set, and denoted K(P ); it consists of The bounded set C points whose orbits under P remain bounded: ˆ sup |P n (z)| < ∞}. K(P ) = {z ∈ C| n
For future reference, let us list in a proposition below the main properties of Julia sets: ˆ →C ˆ be a rational function. Then the following properties Proposition 2.5. Let R : C hold: ˆ which is completely invariant: R−1 (J(R)) = • J(R) is a non-empty compact subset of C J(R); • J(R) = J(Rn ) for all n ∈ N; • J(R) is perfect; ˆ • if J(R) has non-empty interior, then it is the whole of C; ˆ be any open set with U ∩ J(R) 6= ∅. Then there exists n ∈ N such that • let U ⊂ C n R (U) ⊃ J(R); • periodic orbits of R are dense in J(R). Let us further comment on the last property. For a periodic point z0 = Rp (z0 ) of period p its multiplier is the quantity λ = λ(z0 ) = DRp (z0 ). We may speak of the multiplier of a periodic cycle, as it is the same for all points in the cycle by the Chain Rule. In the case when |λ| = 6 1, the dynamics in a sufficiently small neighborhood of the cycle is governed by the Intermediate Value Theorem: when |λ| < 1, the cycle is attracting (super-attracting if λ = 0), if |λ| > 1 it is repelling. Both in the attracting and repelling cases, the dynamics can be locally linearized: (2.2)
ψ(Rp (z)) = λ · ψ(z)
where ψ is a conformal mapping of a small neighborhood of z0 to a disk around 0. By a classical result of Fatou, a rational mapping has at most finitely many non-repelling periodic orbits. Therefore, we may refine the last statement of Proposition 2.5: • repelling periodic orbits are dense in J(R). In the case when λ = e2πiθ , θ ∈ R, the simplest to study is the parabolic case when θ = n/m ∈ Q, so λ is a root of unity. In this case Rp is not locally linearizable; it is not hard to see that z0 ∈ J(R). In the complementary situation, two non-vacuous possibilities are considered: Cremer case, when Rp is not linearizable, and Siegel case, when it is. In the latter case, the linearizing map ψ from (2.2) conjugates the dynamics of Rp on a neighborhood U(z0 ) to the irrational rotation by angle θ (the rotation angle) on a disk around the origin. The maximal such neighborhood of z0 is called a Siegel disk. Siegel disks will prove crucial to our study, and will be discussed in more detail in the next section.
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To conclude the discussion of the basic properties of Julia sets, let us consider the simplest exampes of non-linear rational endomorphisms of the Riemann sphere, the quadratic polynomials. Every affine conjugacy class of quadratic polynomials has a unique representative of the form fc (z) = z 2 + c, the family fc (z) = z 2 + c, c ∈ C
is often referred to as the quadratic family. For a quadratic map the structure of the Julia set is governed by the behaviour of the orbit of the only finite critical point 0. In particular, the following dichotomy holds: Proposition 2.6. Let K = K(fc ) denote the filled Julia set of fc , and J = J(fc ) = ∂K. Then: • 0 ∈ K implies that K is a connected, compact subset of the plane with connected complement; • 0∈ / K implies that K = J is a planar Cantor set. The Mandelbrot set M ⊂ C is defined as the set of parameter values c for which J(fc ) is connected. Continuity of the dependence c 7→ J(fc ). A natural question to pose for polynomials in the quadratic family is whether the Julia set varies continuously with the parameter c. To make sense of this question, recall the definition of the Hausdorff distance distH between compact sets X, Y in the plane (2.1). It turns out that the dependence c 7→ J(fc ) is discontinuous in the Hausdorff distance. For an excellent survey of this problem see the paper of Douady [Do]. The discontinuity which has found most interesting dynamical applications occurs at parameter values for which fc has a parabolic point. We, however, will employ a more obvious discontinuity which is related to Siegel disks. Let us first note that by a result of Douady and Hubbard [DH1] a quadratic polynomial has at most one non-repelling cycle in C. In particular, there is at most one cycle of Siegel disks. Proposition 2.7. Let c∗ ∈ M be a parameter value for which fc has a Siegel disk. Then the map c 7→ J(fc ) is discontinuous at c∗ . Proof. Let z0 be a Siegel periodic point of fc and denote ∆ the Siegel disk around ζ0 , p its period, and θ the rotation angle. By the Implicit Function Theorem, there exists a holomorphic mapping ζ : U(c∗ ) → C such that ζ(c∗) = z0 and ζ(c) is fixed under (fc )p . The mapping ν : c 7→ D(fc )p (ζ(c)) is holomorphic, hence it is either constant or open. If it is constant, all quadratic polynomials have a Siegel disk. This is not possible: for instance, f1/4 has a parabolic fixed point, and thus no other non-repelling cycles. Therefore, ν is open, and in particular, there is a sequence of parameters cn → c∗ such that ζ(cn ) has multiplier e2πipn /qn . Since ζ(cn ) is parabolic, it lies in the Julia set of fcn . Hence distH (J(fc∗ ), J(fcn )) > dist(c∗ , ∂∆)/2
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for n large enough.
Thus an arbitrarily small change of the multiplier of the Siegel point may lead to an implosion of the Siegel disk – its inner radius collapses to zero. Siegel disks of quadratic maps. Let us discuss in more detail the occurence of Siegel disks in the quadratic family. For a number θ ∈ [0, 1) denote [r0 , r1 , . . . , rn , . . .], ri ∈ N ∪ {∞} its possibly finite continued fraction expansion: (2.3)
[r0 , r1 , . . . , rn , . . .] ≡
1
1
r0 +
1
r1 +
1 rn + · · · Such an expansion is defined uniqely if and only if θ ∈ / Q. In this case, the rational convergents pn /qn = [r0 , . . . , rn−1 ] are the closest rational approximants of θ among the numbers with denominators not exceeding qn . In fact, setting λ = e2πiθ , we have ···+
|λh − 1| > |λqn − 1| for all 0 < h < qn+1 , h 6= qn .
The difference |λqn − 1| lies between 2/qn+1 and 2π/qn+1 , therefore the rate of growth of the denominators qn describes how well θ may be approximated with rationals. Definition 2.7. The diophantine numbers of order k, denoted D(k) is the following class of irrationals “badly” approximated by rationals. By definition, θ ∈ D(k) if there exists c > 0 such that qn+1 < cqnk−1 The numbers qn can be calculated from the recurrent relation qn+1 = rn qn + qn−1 , with q0 = 0, q1 = 1. Therefore, θ ∈ D(2) if and only if the sequence {ri } is bounded. Dynamicists call such numbers bounded type (number-theorists prefer constant type). An extreme example of a number of bounded type is the golden mean √ 5−1 θ∗ = = [1, 1, 1, . . .]. 2 The set \ Dk D(2+) ≡ k>2
has full measure in the interval [0, 1). In 1942 Siegel showed:
ˆ of period p. Theorem 2.8 ([Sie]). Let R be an analytic map with an periodic point z0 ∈ C Suppose the multiplier of the cycle λ = e2πiθ with θ ∈ D(2+),
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then the local linearization equation (2.2) holds. The strongest known generalization of this result was proved by Brjuno in 1972: Theorem 2.9 ([Bru]). Suppose (2.4)
B(θ) =
X log(qn+1 ) n
qn
< ∞,
then the conclusion of Siegel’s Theorem holds.
Note that a quadratic polynomial with a fixed Sigel disk with rotation angle θ after an affine change of coordinates can be written as (2.5)
Pθ (z) = z 2 + e2πiθ z.
In 1987 Yoccoz [Yoc] proved the following converse to Brjuno’s Theorem: Theorem 2.10 ([Yoc]). Suppose that for θ ∈ [0, 1) the polynomial Pθ has a Siegel point at the origin. Then B(θ) < ∞. The numbers satisfying (2.4) are called Brjuno numbers; the set of all Brjuno numbers will be denoted B. It is evident that ∪D(k) ⊂ B. The sum of the series (2.4) is called the Brjuno function. For us a different characterization of B will be more useful. Inductively define θ1 = θ and θn+1 = {1/θn }. In this way, θn = [rn−1 , rn , rn+1, . . .].
We define the Yoccoz’s Brjuno function as ∞ X 1 Φ(θ) = θ1 θ2 · · · θn−1 log . θn n=1
One can verify that
B(θ) < ∞ ⇔ Φ(θ) < ∞. The value of the function Φ is related to the size of the Siegel disk in the following way. Definition 2.8. Let P (θ) be a quadratic polynomial with a Siegel disk ∆θ ∋ 0. Consider the conformal isomorphism φ : D 7→ ∆ fixing 0. The conformal radius of the Siegel disk ∆θ is the quantity r(θ) = |φ′ (0)|. For all other θ ∈ [0, ∞) we set r(θ) = 0. By the Koebe One-Quarter Theorem of classical complex analysis, the internal radius of ∆θ is at least r(θ)/4. Yoccoz [Yoc] has shown that the sum Φ(θ) + log r(θ) is bounded below independently of θ ∈ B. Recently, Buff and Ch´eritat have greatly improved this result by showing that:
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Theorem 2.11 ([BC2]). The function θ 7→ Φ(θ) + log r(θ) extends to R as a 1-periodic continuous function. We remark that the following stronger conjecture exists (see [MMY]): Marmi-Moussa-Yoccoz Conjecture. [MMY] The function θ 7→ Φ(θ)+log r(θ) is H¨older of exponent 1/2. Dependence of the conformal radius of a Siegel disk on the paramater. In this section we will show that the conformal radius of a Siegel disk varies continuously with the Julia set. To that end we will need a preliminary definition: Definition 2.9. Let (Un , un ) be a sequence of topological disks Un ⊂ C with marked points un ∈ Un . The kernel or Carath´eodory convergence (Un , un ) → (U, u) means the following: • un → u; • for any compact K ⊂ U and for all n sufficiently large, K ⊂ Un ; • for any open connected set W ∋ u, if W ⊂ Un for infinitely many n, then W ⊂ U. The topology on the set of pointed domains which corresponds to the above definition of convergence is again called kernel or Carath´eodory topology. The meaning of this topology is as follows. For a pointed domain (U, u) denote φ(U,u) : D → U
the unique conformal isomorphism with φ(U,u) (0) = u, and (φ(U,u) )′ (0) > 0. By the Riemann Mapping Theorem, the correspondence ι : (U, u) 7→ φ(U,u) establishes a bijection between marked topological disks properly contained in C and univalent maps φ : D → C with φ′ (0) > 0. The following theorem is due to Carath´eodory, a proof may be found in [Pom]: Theorem 2.12 (Carath´ eodory Kernel Theorem). The mapping ι is a homeomorphism with respect to the Carath´eodory topology on domains and the compact-open topology on maps. Proposition 2.13. The conformal radius of a quadratic Siegel disk varies continuously with respect to the Hausdorff distance on Julia sets. Proof. To fix the ideas, consider the family Pθ with θ ∈ B and denote ∆θ the Siegel disk of Pθ . It is easy to see that the Hausdorff convergence J(Pθn ) → J(Pθ ) implies the Carath´eodory convergence of the pointed domains (∆θn , 0) → (∆, 0). The proposition follows from this and the Carath´eodory Kernel Theorem.
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Non-computability of the Yoccoz’s Brjuno function. By the analysis similar to our proof of the Main Theorem, we can also prove a direct non-computability result for the Yoccoz’s Brjuno function Φ: Theorem 2.14 (Non-computability of Φ). There exists a parameter value θ ∈ R/Z such that Φ(θ) is not computable by any oracle Turing Machine with access to θ. It is worth noting that Marmi-Moussa-Yoccoz Conjecture as stated above and Theorem 2.14 imply the existence of a non-computable quadratic Julia set. To see this, we first formulate: Conditional Implication . If Marmi-Moussa-Yoccoz Conjecture holds, then the function ν : θ 7→ Φ(θ) + log r(θ) is computable by one Turing Machine on the entire interval [0, 1]. We use the following result of Buff and Ch´eritat ([BC2]). Lemma 2.15 ([BC2]). For any rational point θ =
p q
∈ [0, 1] denote, as before,
Pθ (z) = e2πiθ z + z 2 , and let the Taylor expansion of Pθ◦q (z) at 0 start with Pθ◦q (z) = z + Az q+1 + . . . , for q ∈ N 1/q 1 . Denote by Φtrunc the modification of Φ applied to rational numbers Let L(θ) = qA where the sum is truncated before the infinite term. Then we have the following explicit formula for computing ν(θ): (2.6)
ν(θ) = Φtrunc (θ) + log L(θ) +
log 2π . q
Equation (2.6) allows us to compute the value of ν easily at every rational θ ∈ Q∩[0, 1] with an arbitrarily good precision. In addition, assuming the conjecture, we have |ν(x)−ν(y)| < 2−n whenever |x − y| < c · 2−2n for some constant c, hence ν has an (easily) computable modulus of continuity. These two facts together imply that ν is computable by a single machine of the interval [0, 1] (see for example Proposition 2.6 in [Ko2]). This implies the Conditional Implication. The following conditional result follows: Lemma 2.16 (Conditional). Suppose the Conditional Implication holds. Let θ ∈ [0, 1] be such that Φ(θ) is finite. Then there is an oracle Turing Machine M1φ computing Φ(θ) with an oracle access to θ if and only if there is an oracle Turing Machine M2φ computing r(θ) with an oracle access to θ.
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Proof. Suppose that M1φ computes Φ(θ) for some θ. Let M φ be the machine uniformly computing the function ν. Then we can use M1φ and M φ to compute log r(θ) = ν(θ) − Φ(θ) with an arbitrarily good precision. We can then use this construction to give a machine M2φ which computes r(θ). The opposite direction is proved analogously. Lemma 2.16 with Theorem 2.14 imply that there is a θ for which r(θ) is non-computable. Considerations of the Riemann Mapping imply that for this value of θ the Julia set of Pθ is also non-computable. Note that for the proof of Conditional Implication we did not need the full power of the conjecture. All we needed is some computable bound on the modulus of continuity of ν. Outline of the construction of a non-computable quadratic Julia set. We are now prepared to outline the idea of our construction. The outline given below is rather rough and suffers from obvious logical deficiencies. However, it captures the idea of the proof in a simple to understand form. Suppose that every Julia set of a polynomial Pθ is computable by an oracle machine M φ , where φ represents θ. There are countably many machines, so we can enumerate them M1φ , M2φ , . . .. Denote by Si the domain on which Miφ computes JPθ properly. Then we must have: S (1) C = ∞ i=1 Si , (2) for each i, the function J : θ 7→ J(Pθ ) is continuous on Si .
Figure 1. The Siegel disks of Pθ for θ given by the continued fractions [1, 1, 1, . . .], [1, 1, 1, 20, 1, . . .], and [1, 1, 1, 20, 1, 1, 1, 30, 1, . . .] Let us start with a machine Mnφ1 which computes J(Pθ∗ ) for θ∗ = [1, 1, 1, . . .]. If any of the digits ri in this infinite continued fraction is changed to a sufficiently large N ∈ N,
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the conformal radius of the Siegel disk will become small. For N → ∞ the Siegel disk will implode and its center will become a parabolic fixed point in the Julia set. Given the continuity of the dependence of the conformal radius of the Siegel disk on the Julia set, we have the following: There exists i1 > 1 such that for every θ1 whose continued fraction starts with i1 ones, for the Julia set of Pθ1 to be computable by Mnφ1 , it must possess a Siegel disk of a conformal radius r(θ1 ) > r(θ∗ )(1 − 1/8). We can thus “fool” the machine Mnφ1 by selecting θ1 given by a continued fraction where all digits are ones except ri1 = N1 >> 1. If we are careful, we can do it so that (2.7)
r(θ∗ )(1 − 1/4) < r(θ1 ) < r(θ∗ )(1 − 1/8).
To “fool” the machine Mnφ2 we then change a digit ri2 for i2 > i1 sufficiently far in the continued fraction of θ1 to a large N2 . In this way, we will obtain a Brjuno number θ2 for which (2.8)
r(θ∗ )(1 − 1/4 − 1/8) < r(θ2 ) < r(θ∗ )(1 − 1/4).
Continuing in this manner we will arrive at a limiting Brjuno number θ∞ for which the Julia set is uncomputable. To make such a scheme work, we need a careful analysis of the dependence of the conformal radius on the parameter. In this a key role is played by Theorem 2.11 of Buff and Ch´eritat which allows us to obtain a controlled change in the value of r(α) by changing Φ(α). The relevant analysis is carried out in the next section. A note on the connection with [BC1]. A. Ch´eritat has pointed to us that methods of [BC1], where Siegel disks with smooth boundaries are constructed for the quadratic family can be used to derive the Main Theorem. We discuss this in the section following the proofs of the main theorems. We note here that the argument we give is based on quite elementary estimates of the function Φ and is thus accessible to non-dynamicists. It has an added advantage of yielding Theorem 2.14. 3. Making Small Changes to Φ 3.1. Small Changes to Φ. A key step of the construction outlined above is making careful adjustments of r(θi ) as in the first two steps (2.7) and (2.8) above. We do not have a direct control over the value of r(α), but Buff and Ch´eritat’s Theorem 2.11 shows that small decreases of r(α) we would like to make correspond to a small controlled increment of the value of Φ(α). Estimates of a similar nature has appeared in the works of various authors (compare, for example, with [BC1]). For a number γ = [a1 , a2 , . . .] ∈ R \ Q we denote 1 αi (γ) = , 1 ai + 1 ai+1 + ai+2 + . . .
NON-COMPUTABLE JULIA SETS
so that Φ(γ) =
X
α1 (γ)α2 (γ) . . . αn−1 (γ) log
n≥1
13
1 . αn (γ)
The main goal of this section is to prove the following lemma: Lemma 3.1. For any initial segment I = (a0 , a1 , . . . , an ), write α = [a0 , a1 , . . . , an , 1, 1, 1, . . . ]. Then for any ε > 0, there is an m > 0 and an integer N such that if we write β = [a0 , a1 , . . . , an , 1, 1, . . . , 1, N, 1, 1, . . .], where the N is located in the n + m-th position, and Φ(α) + ε < Φ(β) < Φ(α) + 2ε. The proof is technical and will require some preparation. The idea is to choose an m large enough, so that changing an+m (which will eventually be N) by 1 changes the value of Φ by a very small amount (< ε). When N → ∞, Φ(α) → ∞, hence the value of Φ must hit the interval (Φ(α) + ε, Φ(α) + 2ε). Denote 1 Φ− (α) = Φ(α) − α0 α1 . . . αn+m−1 log . αm+n The value of the integer m > 0 is yet to be determined. Denote β N = [a0 , a1 , . . . , an , 1, 1, . . . , 1, N, 1, 1, . . .]. We prove the following Lemma 3.2. For any N and i ≤ n + m we have N log αi (β ) < 2i−(n+m) . N +1 αi (β )
Proof. We proove the lemma by induction on i, starting from the base case i = n + m, and proceeding down to i = 0. The base case is i = n + m, we want to prove N α (β ) n+m < 1. log αn+m (β N +1 ) We have
rn+m
1 N +x = N +1+x = 1+ 1 , 1 N +x N +x N +1+x < 2, and | log rn+m | < 1.
αn+m (β N ) = = αn+m (β N +1 )
for some x < 1. Hence 1 < rn+m
Induction step. Supposing that the statement is true for i + 1, we prove it for i. We have 1 N ai + αi+1 (β N +1) αi (β ) ai + αi+1 (β N ) = = . 1 αi (β N +1 ) ai + αi+1 (β N ) ai + αi+1 (β N +1 )
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M. BRAVERMAN, M. YAMPOLSKY
Suppose that αi+1 (β N +1 ) ≥ αi+1 (β N ). Then we know that αi+1 (β N +1 ) i+1−(n+m) < e2 , N αi+1 (β )
and we want to prove that ai + αi+1 (β N +1 ) i−(n+m) < e2 , N ai + αi+1 (β ) since this expression is obviously bigger than 1. The situation would have been the same if αi+1 (β N +1 ) ≤ αi+1 (β N ), with the numerator and the denominator exchanged. In other words, it is enough to prove that for 0 < d < c < 1 and a pair of integers r ≥ 0 and k ≥ 1, k+c c −r −r−1 < e2 implies that < e2 . d k+d First of all, it is easy to see that for k ≥ 1, k+c 1+c ≤ , k+d 1+d hence it suffices to show that −r 1/2 1+c −r−1 < e2 = e2 . 1+d Thus we need to demonstrate that 1 + c c 1/2 . < 1+d d This is equivalent to (1 + 2c + c2 )d < (1 + 2d + d2 )c ⇔ d + c2 d < c + d2 c ⇔ cd(c − d) < c − d.
The last inequality holds, since cd < 1 and c − d > 0.
The following lemma is proven exactly as the previous one with a shifted induction base. Lemma 3.3. For any N and i < n + m we have N α (β ) i log < 2i−(n+m) . αi (β 1 )
Proof. The proof goes by induction exactly as before. We need to verify the base case i = n + m − 1. For this value of i, √ 5−1 1 , αn+m−1 (β 1 ) = = φ 2 and 1 1 < 1. ≤ αn+m−1 (β N ) = 1 φ 1 + N +...
NON-COMPUTABLE JULIA SETS
Hence we have
N α (β ) n+m−1 < log φ ≈ 0.481 < 0.5 = 2−1 . log 1 αn+m−1 (β )
We now bound the influence of the change on the log
15
1 terms. αi
Lemma 3.4. For any N and i < n + m, 1 log N) α (β i < 2i−(n+m)+1 . log 1 log N +1 α (β ) i
N
Proof. Assume that αi (β ) ≤ αi (β case we need to prove
N +1
), the reverse case is done in the same way. In this
1 αi (β N ) i−(n+m)+1 < e2 . 1 log αi (β N +1 ) log
1 and αi (β N +1 ) = Denote c = αi+1 (β N ) and d = αi+1 (β N +1 ). Then we have αi (β N ) = k+c 1 for some integer k ≥ 1. Hence 0 < d ≤ c < 1. We have k+d 1 log log(k + c) αi (β N ) . = 1 log(k + d) log αi (β N +1 ) c log(k + c) c i−(n+m)+1 By lemma 3.2 we know that < e2 , hence it suffices to show that ≤ . d log(k + d) d log(k + d) log(k + x) log(k + c) ≤ . Consider the function f (x) = on This is equivalent to c d x the interval (0, 1). The reader can readily verify that f ′ (x) < 0 for x ∈ (0, 1) so that f is decreasing on this interval, and hence f (c) ≤ f (d), which completes the proof. In the same way that Lemma 3.4 follows from Lemma 3.2, the following Lemma follows from Lemma 3.3 Lemma 3.5. For any N and i < n + m − 1, 1 log N) α (β i log < 2i−(n+m)+1 . 1 log 1 αi (β )
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M. BRAVERMAN, M. YAMPOLSKY
Proof. As in lemma 3.4.
We are now ready to bound the influence of changes in N on the value of Φ− . Lemma 3.6. For any α of the form as in lemma 3.1 and for any ε > 0, there is an m0 > 0 such that for any N and any m ≥ m0 , ε |Φ− (β N ) − Φ− (β 1 )| < . 4 P 1 Proof. The in the expression for Φ(β ) converges, hence there is an m1 > 1 such that the P 1 ε tail of the sum i≥n+m1 α1 α2 . . . αi−1 log < . We will show how to choose m0 ≥ m1 αi 16 to satisfy the conclusion of the lemma. We bound the influence of the change from β 1 to β N using lemmas 3.3 and 3.5. The influence on each of the “head elements” (i < n + m1 ) is bounded by 1 1 1 i−1 α (β ) . . . α (β ) log 1 i−1 1 αi (β ) X j−(n+m) log 2 + 2i−(n+m)+1 < 2i−(n+m)+2 < 2m1 +2−m . < 1 j=1 α1 (β N ) . . . αi−1 (β N ) log αi (β N )
By making m sufficiently large (i.e. by choosing a sufficiently large m0 ) we can ensure that 1 α1 (β N ) . . . αi−1 (β N ) log ε ε αi (β N ) 1− n + m are not affected by the change, and the change decreases αn+m , so that αn+m (β N ) ≤ αn+m (β 1 ). Hence 1 α1 (β N ) . . . αi−1 (β N ) log α1 (β N ) . . . αn+m (β N ) αi (β N ) = log ≤ log 1 α1 (β 1 ) . . . αn+m (β 1 ) α1 (β 1 ) . . . αi−1 (β 1 ) log αi (β 1 ) log
n+m−1 X α1 (β N ) . . . αn+m−1 (β N ) < 2j−(n+m) < 1 α1 (β 1 ) . . . αn+m−1 (β 1 ) j=1
So in this case each term could increase by a factor of e at most. We see that after the change each term of the tail could increase by a factor of e at most. eε The value of the tail remains positive in the interval (0, ], hence the change in the tail 16 3ε eε < . is bounded by 16 16 So the total change in Φ− is bounded by ε 3ε ε change in the “head” + change in the “tail” < + = . 16 16 4 The following Lemma follows immediately from Lemma 3.6.
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M. BRAVERMAN, M. YAMPOLSKY
Lemma 3.7. For any ε and for the same m0 (ε) as in lemma 3.6, for any m ≥ m0 and N, ε |Φ− (β N ) − Φ− (β N +1 )| < . 2 Proof. We have ε ε ε |Φ− (β N ) − Φ− (β N +1 )| ≤ |Φ− (β N ) − Φ− (β 1 )| + |Φ− (β 1) − Φ− (β N +1 )| < + = . 4 4 2 1 We will now have to take a closer look at the term α1 . . . αn+m−1 log = Φ(α)−Φ− (α). αm+m For this we will need a slightly sharper version of Lemma 3.2. Lemma 3.8. For any N and i ≤ n + m we have N 2i−(n+m) α (β ) i < log . αi (β N +1 ) N
Proof. The induction works exactly as in Lemma 3.2, we only need to verify the base case i = n + m. We have 1 N αn+m (β ) N +1+φ 1 N +φ = = =1+ , N +1 1 αn+m (β ) N +φ N +φ N +1+φ hence 1< So
1 αn+m (β N ) 1, αk−1 αk < . 2 Proof. There is an integer l ≥ 1 such that 1 1 1 αk < αk = . αk−1 αk = l + αk αk + αk 2 Denote Φ1 (α) = α1 . . . αn+m−1 log following.
1 αn+m
= Φ(α) − Φ− (α), we are now ready to prove the
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19
Lemma 3.10. For sufficiently large m, for any N, ε Φ1 (β N +1 ) − Φ1 (β N ) < . 2 Proof. According to Lemma 3.8 we have n+m−1 N +1 N +1 X α (β ) . . . α (β ) 1 1 n+m−1 i−(n+m) log < 2 /N < . α1 (β N ) . . . αn+m−1 (β N ) N i=1
Hence α1 (β N +1 ) . . . αn+m−1 (β N +1 ) < α1 (β N ) . . . αn+m−1 (β N )e1/N , and 1 log log(N + 1 + φ) αn+m (β N +1 ) = Φ1 (β N )e1/N . Φ1 (β N +1 ) < Φ1 (β N )e1/N 1 log(N + φ) log αn+m (β N )
Hence
1/N log(N + 1 + 1/φ) −1 < Φ (β ) − Φ (β ) < Φ (β ) e log(N + 1/φ) e log(N + 1 + 1/φ) 1 N Φ (β ) (1 + ) −1 . N log(N + 1/φ) log(N + 1 + 1/φ) , then (N + 1/φ)x = We make the following calcualtions. Denote x = log(N + 1/φ) N + 1 + 1/φ, and 1 1 < e N + 1/φ . (N + 1/φ)x−1 = 1 + N + 1/φ 3 3 3 N + 1/φ > e1/3 , and so x − 1 < < , thus x < 1 + . N + 1/φ N N 1
N +1
1
N
1
N
By Lemma 3.9 we have 1 Φ (β ) = α1 (β ) . . . αn+m−1 (β ) log < αn+m (β N ) 1
N
N
N
Thus
(n+m−1)/2 1 log(N + 1/φ). 2
e log(N + 1 + 1/φ) Φ (β ) − Φ (β ) < Φ (β ) (1 + ) −1 < N log(N + 1/φ) (n+m−1)/2 (n+m−1)/2 14 1 1 log(N + 1/φ) ((1 + e/N)(1 + 3/N) − 1) < log(N + 1/φ) . 2 2 N 14 ε Since ∈ o(1/ log(N + 1/φ)), this expression can be always made less than by choosing N 2 m large enough. 1
N +1
1
N
1
N
Lemmas 3.7 and 3.10 yield the following
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M. BRAVERMAN, M. YAMPOLSKY
Lemma 3.11. For sufficiently large m, for any N, Φ(β N +1 ) − Φ(β N ) < ε. Proof. We use Lemmas 3.7 and 3.10. For sufficiently large m, Φ(β N +1 ) − Φ(β N ) ≤ Φ− (β N +1 ) − Φ− (β N ) + Φ1 (β N +1 ) − Φ1 (β N )
and
1 · α1 (β 1 ) . . . αn+m−1 (β 1 ) e
1 log(N + 1/φ) 1 1 · Φ (β ). e log(1 + 1/φ) The latter expression obviously goes to ∞ as N → ∞. Φ1 (β N ) >
We are now ready to prove Lemma 3.1.
Proof. (of Lemma 3.1). Choose m large enough for lemma 3.11 to hold. Increase N by one at a time starting with N = 1. We know that Φ(β 1 ) = Φ(α) < Φ(α) + ε, and by Lemma 3.12, there exists an M with Φ(β M ) > Φ(α) + ε. Let N be the smallest such M. Then Φ(β N −1 ) ≤ Φ(α) + ε, and by Lemma 3.11 Φ(β N ) < Φ(β N −1 ) + ε ≤ Φ(α) + 2ε.
Hence Choosing β = β N
Φ(α) + ε < Φ(β N ) < Φ(α) + 2ε. completes the proof.
The following is an important note on the proof which we will use later in the section. Lemma 3.13. After the process described above denote the total variation by X 1 1 N N N 1 1 1 α1 (β )α2 (β ) . . . αi−1 (β ) log . T = − α (β )α (β ) . . . α (β ) log 1 2 i−1 αi (β N ) αi (β 1 ) i≥0
Then T is finite, and satisfies T ≤ 3ε.
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21
Proof. Note that m > m1 . We split T into two parts: the variation T1 on the first n+m1 −1 terms of the sum, and the variation T2 on the rest of the terms (see proof of lemma 3.6 for the definition of m1 ). Note that T = T1 + T2 . By the first part of the proof of lemma 3.6 ε we immediately see that T1 < . 16 P 1 < We also know (by the definition of m1 ), that i≥n+m1 α1 (β 1 )α2 (β 1 ) . . . αi−1 (β 1 ) log αi (β 1 ) ε . Denote the tolal increase of the second part by T2+ ≥ 0, and the total decrease by 16− T2 ≥ 0. Then T2 = T2+ + T2− . Note that the total increase of the formula is bounded ε 1 by 2ε, and the decrease in the first part is less than , hence T2+ − T2− < 2 ε. On 16 16 2 ε + − . So T2 < 2 ε, and the other hand, all the terms are nonnegative, hence T2 < 16 16 3 ε 3 + − < 3ε, which completes the proof. T2 = T2 + T2 < 2 ε. Finally, T = T1 + T2 < 2 ε + 16 16 16 The following Lemma follows by the same argument as Lemma 3.6 by taking N ≥ 1 to be an arbitrary real number, not necessairily an integer. Lemma 3.14. For a β 1 as above, for any ε > 0 there is an m0 > 0, such that for any m ≥ m0 , and for any tail I = [an+m , an+m+1 , . . .] if we denote β I = [a1 , a2 , . . . , an , 1, 1, . . . , 1, an+m , an+m+1 , . . .],
then
X
α1 (β 1 )α2 (β 1 ) . . . αi−1 (β 1 ) log
i≥n+m
and
1 < ε, αi (β 1 )
n+m−1 X i=1
α1 (β I ) . . . αi−1 (β I ) log 1 − α1 (β 1 ) . . . αi−1 (β 1 ) log 1 < ε. αi (β I ) αi (β 1 )
We will also need the following Lemma in the proof of the Main Theorem.
Lemma 3.15. Let α = [a0 , a1 , a2 , . . .] and let ε > 0 be given. Then there is an N = N(ε) such that for any n ≥ N we have Φ(αn ) < Φ(α)+ε, where αn = [a0 , a1 , . . . , an−1 , 1, 1, 1, . . .]. The proof is not hard and is similar to the proof of Lemma 3.6. The influence on the “head” is small because the switch to 1’s occurs far away, and the “tail” does not increase too much because it can be shown to be very small (< 4ε ) after the change with n large enough. We leave the details to the reader. 4. Proving the Main Theorem We are now ready to prove the Main Theorem.
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M. BRAVERMAN, M. YAMPOLSKY
Theorem 4.1. There exists a parameter value θ ∈ R/Z such that the Julia set of the quadratic polynomial fθ (z) = z 2 + e2πiθ z is not computable. We note the the Main Theorem follows from Theorem 4.1 in an elementary way. Indeed, let θ ∈ R/Z be as promiced by Theorem 4.1. Then given an oracle access to θ it is easy to compute λ = e2πiθ , and vice versa. This implies that we could simulate the oracle for λ given an oracle for θ, and thus it is impossible to compute J(z 2 + λz) with an oracle access to λ. Setting c = λ2 /4 − λ/2 and Tc (z) = z − 2λ, we have J(z 2 + λz) = Tc (J(z 2 + c)).
Hence being able to compute J(z 2 +c) with an oracle access to c would allow us to compute J(z 2 + λz) with an oracle access to λ. Proof of Theorem 4.1. There are countably many oracle Turing Machines, hence we can enumerate them as M1φ , M2φ , . . .. We prove the theorem by constructing a number c that “fools” all the oracle Turing Machines M1φ , M2φ , . . . which are trying to compute Jc = J(fc ). We proceed iteratively, so on step i we “fool” the first i machines Mkφ . We then take the process to the limit, obtaining a number on which none of the machines works. Recall that r(θ) is the conformal radius of the Siegel disk associated with the polynomial Pθ (z) = z 2 + e2πiθ z, or zero, if θ is not a Brjuno number. On each iteration i we maintain an initial segment Ii = [a0 , a1 , . . . , aNi ] and an interval [li , ri ] such that the following properties are maintained: (4.1)
ri = r(γi ), where γi = [Ii , 1, 1, . . .],
and (4.2)
For any β = [Ii , tNi +1 , tNi +2 , . . .] with r(β) ∈ [li , ri ], the machines M1φ , M2φ , . . . , Miφ fail on β.
Moreover, the intervals we construct are nested: [li , ri ] ⊂ [li−1 , ri−1 ], and the sequence Ii contains Ii−1 as the initial segment. We also keep track of the variation (4.3) X 1 1 α1 (γi+1 )α2 (γi+1 ) . . . αj−1 (γi+1 ) log Ti = − α (γ )α (γ ) . . . α (γ ) log 1 i 2 i j−1 i α (γ ) α (γ ) j i+1 j i i≥0
at each step of the process, in order to be able to pass to the limiting sequence [a1 , a2 , a3 , . . .] in the end. For the basis of induction, set I0 = [1], r0 = r(γ0 ) and l0 = r0 /2, where γ0 = [1, 1, 1, . . .]. Then for i = 0 condition (4.1) holds by definition and condition (4.2) holds because it is empty.
NON-COMPUTABLE JULIA SETS
23
The induction step. We now have the conditions (4.1) and (4.2) for some i and would like to extend them to i + 1. φ Consider the machine M = Mi+1 . Denote by S the set of points β = [Ii , tNi +1 , tNi +2 , . . .] with r(β) ∈ [li , ri ] on which M works correctly. Let R be the set of possible conformal radii on S. There are two possibilities: Case 1: There exist ε0 > 0 and m0 ∈ N such that for every β ∈ S of the form β = [Ii , 1, 1, . . . , 1, . . .] we have |ri − r(β)| > ε0 . | {z } m0
In this case, select 0 < ε ≤ ε0 such that ri − ε > li . Set
Ii+1 = [Ii , 1, 1, . . . , 1], li+1 = ri − ε, and ri+1 = ri . | {z } m0
Since γi+1 = [Ii , 1, 1, . . .] = γi , we have r(γi+1 ) = r(γi) = ri = ri+1 and the condition (4.1) is satisfied. Suppose β = [Ii+1 , tNi+1 +1 , tNi+1 +2 , . . .] with r(β) ∈ [li+1 , ri+1 ]. Ii+1 is an extension of Ii , φ and the machines M1φ , M2φ , . . . , Miφ fail on β by the induction assumption. Mi+1 fails on β because r(β) ∈ [li+1 , ri+1 ], and so r(β) ∈ / R. This shows (4.2) and completes the proof for this case. Note that in this case we have γi+1 = γi, and so Ti = 0. The complementary case is the main part of the argument: Case 2. For every ε > 0 and m ∈ N we can find β ∈ S starting with Ii followed by m ones so that ri − ε < r(β) ≤ ri
(4.4)
Choose the number m of 1’s to be such that Lemma 3.14 holds with ε = 2−i . Choose an ε such that ri − 3ε > li , and let β ∈ S as above satisfying ri − ε < r(β) ≤ ri . Denote log(ri − ε) − log(ri − 2ε) log(ri − 2ε) − log(ri − 3ε) −i ε0 = min > 0. , ,2 8 8
Theorem 2.11 says that θ 7→ Φ(θ) + log r(θ) extends to a continuous 1-periodic function on R. Denote this function by f (θ). f is continuous and periodic, hence it must be uniformly continuous, and there is a δ > 0 such that if |x − y| < δ then |f (x) − f (y)| < ε0 . By Theorem 2.4 the Julia sets of Pθ depend continuously on θ with respect to the Hausdorff metric for θ ∈ S. By Proposition 2.13 the conformal radius r(•) is continuous on S. Hence there is a δ0 > 0 such that |r(x) − r(β)| < ε whenever |x − β| < δ0 and x ∈ S.
We choose m large enough, so that for any ζ = [Ii , 1, 1, . . . , 1, . . .], |γi − ζ| < δ. | {z } m
Write β = [Ii , 1, 1, . . . , 1, tNi +1 , tNi +2 , . . .]. By Lemma 3.15, there is an N such that for | {z } m
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M. BRAVERMAN, M. YAMPOLSKY
any n ≥ N, βn = [Ii , 1, 1, . . . , 1, tNi +1 , . . . , tNi +n , 1, 1, . . .] satisfies Φ(βn ) < Φ(β) + ε0 . | {z } m
We can choose n ≥ N large enough so that for any x with the initial segment Ii′ = [Ii , 1, 1, . . . , 1, tNi +1 , tNi +2 , . . . , tNi +n ], |x − β| < δ/2 and |x − β| < δ0 . | {z } m
Start with ω0 = βn = [Ii′ , 1, 1, . . .]. We have |ω0 − β| < δ, hence |f (ω0) − f (β)| < ε0 . So
log r(ω0) = f (ω0) − Φ(ω0 ) > f (β) − ε0 − Φ(β) − ε0 = log r(β) − 2ε0 . By Lemma 3.1 we can extend ω0 to ω1 = [Ii1 , 1, 1, . . .] so that Φ(ω0 ) + 2ε0 < Φ(ω1 ) < Φ(ω0 ) + 4ε0 . Hence and
log(r(ω1 )) = f (ω1 ) − Φ(ω1 ) > f (ω0 ) − ε0 − Φ(ω0 ) − 4ε0 = log(r(ω0)) − 5ε0 , log(r(ω1 )) = f (ω1 ) − Φ(ω1 ) < f (ω0 ) + ε0 − Φ(ω0 ) − 2ε0 = log(r(ω0 )) − ε0 .
Hence log(r(ω0 )) − 5ε0 < log(r(ω1 )) < log(r(ω0 )) − ε0 . In the same fasion we can extend Ii1 to Ii2 , and obtain ω2 = [Ii2 , 1, 1, . . .] so that log(r(ω1)) − 5ε0 < log(r(ω2 )) < log(r(ω1 )) − ε0 . Recall that log(r(ω0 )) > log(r(β)) − 2ε0 > log(ri − ε) − 2ε0 ≥ log(ri − 2ε) + 6ε0 . Hence after finitely many steps we will obtain Iik and ωk = [Iik , 1, 1, . . .] such that log(ri − 3ε) + ε0 < log(r(ωk )) < log(ri − 3ε) + 6ε0 < log(ri − 2ε).
Choose Ii+1 = Iik , γi+1 = ωk , li+1 = li , and ri+1 = r(ωk ). We have li+1 < ri+1 < ri − 2ε. Condition (4.1) is satisfied by definition. Condition (4.2) is satisfied for the first i machines because [li+1 , ri+1 ] is a subinterval of [li , ri ], and Ii+1 is an extension of Ii . Condition (4.2) is satisfied fot the i + 1-st machine M, because M could work correctly on x = [Ii′ , . . .] only with r(x) ∈ [ri − 2ε, ri + ε], and [li+1 , ri+1 ] is disjoint from [ri − 2ε, ri + ε]. We now estimate the total variation Ti in the process described above. By Lemma 3.14 we know that X 1 α1 (γi)α2 (γi ) . . . αj−1 (γi ) log < 2−i , αj (γi ) j≥N +1 i
and
Ni X 1 1 α1 (γi+1 ) . . . αj−1 (γi+1 ) log < 2−i . − α1 (γi ) . . . αj−1 (γi ) log αj (γi+1 ) αj (γi ) j=1
Denote d = Φ(γi+1 ) − Φ(γi ). Refer to the first Ni terms as the “head terms”, and the rest of the term as the “tail terms”. Denote the total increase in head terms from γi to γi+1 by h+ , and the total decrease by h− . Similarly define t+ and t− . Then d = h+ − h− + t+ − t− , and Ti = h+ + h− + t+ + t− . By the second inequality we have h+ + h− < 2−i . By the first one we have t− < 2−i . Hence t+ ≤ d + h+ + h− + t− < d + 2 · 2−i ,
NON-COMPUTABLE JULIA SETS
25
and Ti < 2−i + 2−i + d + 2 · 2−i = d + 4 · 2−i . We now bound d. d = Φ(γi+1 ) − Φ(γi ) = f (γi+1 ) − log(r(γi+1)) − f (γi) + log(r(γi )) < ε0 + log(r(γi )) − log(r(γi+1 )) < 2−i + log(ri ) − log(ri+1 ).
Hence Ti < log(ri ) − log(ri+1 ) + 5 · 2−i . Overall we get a bound on the sum of all the Ti ’s: ∞ X i=0
Ti < log(r0 ) − log(l0 ) + 5 ·
∞ X i=1
2−i = log(r0 ) − log(l0 ) + 10 < ∞.
Passing to the Limit. The completion of the proof relies on the following Lemma: Lemma 4.2. Let Ii be the initial segment of γi as above. Then the sequence {Φ(γi )} converges to a limit L = lim Φ(γi ). i→∞
Denote by I the inductive limit of the segments above, and let γ be the number represented by I, then Φ(γ) = L. 1 , and bi = α1 (γ)α2 (γ) . . . αi−1 (γ) log αi1(γ) . Proof. Denote bij = α1 (γj )α2 (γj ) . . . αi−1 (γj ) log αi (γ j) Our first claim is that for any i, limj→∞ bij = bi . Note that on step j the position being modified is at least j, hence by Lemmas 3.3 and 3.5 we see that for any j > i, 1 log α (γ ) αi (γj ) log i j < 2i−j , and log < 2i−j+1 . 1 αi (γ) log αi (γ)
Hence for all j > i, i−1 X k−j log bij < 2 + 2i−j+1 < 2i−j + 2i−j+1 < 2i−j+2. bi k=1 bi,i+2 bij Hence limj→∞ bi = 1. We know that bi is finite since log bi < 1, and bi,i+2 is finite by the definition of Φ(γi+2 ). Finally, limj→∞ bij = bi . P P ∞ ∞ Denote aij = bi,j+1 − bij and ai0 = bi1 . Then i=1 |aij | is the sum of all the j=1 variations T discussed above, and has been shown to be bounded by a fixed constant. Hence P∞ P∞ P∞ P∞ i i=1 |aij | < ∞. Hence the sum converges absolutely. j=1 i=1 |aij | = Φ(γ1 ) + j=0 Pj−1 P We have k=0 aik = bij . Taking j → ∞ yields ∞ k=0 aik = limj→∞ bij = bi . Hence Φ(γ) =
∞ X i=1
bi =
∞ X ∞ X i=1 k=0
aik .
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M. BRAVERMAN, M. YAMPOLSKY
For any j, P both Φ(γj ) and Φ(γ ) are finite, hence Φ(γj+1 ) − Φ(γj ) = P Pj+1 ∞ ∞ ∞ b = (b − b ) = ij i=1 ij i=1 i,j+1 i=1 aij . Hence L = lim Φ(γj ) = Φ(γ1 ) + j→∞
∞ X j=1
(Φ(γj+1) − Φ(γj )) =
∞ X
bi1 +
i=1
∞ X ∞ X
aij =
j=1 i=1
P∞
i=1 bi,j+1
∞ X ∞ X
−
aij .
j=0 i=1
P P∞ By the absolute convergence of ∞ i=1 |aij | < ∞, the limit L above exists, and we can j=0 apply Fubini’s theorem to obtain ∞ ∞ X ∞ ∞ X X X aij = aij = L, Φ(γ) = i=1 j=0
j=0 i=1
which completes the proof.
Finalizing the argument. Let γ be the limit from the previous section. We claim the no machine works properly on γ. For any i, Ii is an initial segment of γ by definition. We only need to see that r(γ) ∈ [li , ri ] to show that Miφ fails on γ. We claim that (4.5)
r(γ) = lim ri . i→∞
r(γ) ∈ [li , ri ] follows, since li is nondecreasing and ri is nonincreasing. To show (4.5) observe that γ = limi→∞ γi , and by continuity of f , f (γ) = limi→∞ f (γi). Together with Lemma 4.2 we obtain lim log(ri ) = lim (f (γi ) − Φ(γi )) = lim f (γi ) − lim Φ(γi ) = f (γ) − Φ(γ) = log(r(γ)),
i→∞
i→∞
i→∞
i→∞
thus (4.5) follows, which completes the proof. Non-computability of Yoccoz’s Brjuno function. The proof of Theorem 2.14 requires very minor modifications to the preceding argument. We proceed by induction maintaining on iteration i an initial segment Ii and an interval [li , ri ] such that the following analogues of (4.1) and (4.2) are maintained: (4.6)
li = Φ(γi ), where γi = [Ii , 1, 1, . . .],
and (4.7)
For any β = [Ii , tNi +1 , tNi +2 , . . .] with Φ(β) ∈ [li , ri ],
the machines M1φ , M2φ , . . . , Miφ fail on β. The induction step is carried out similarly to the proof of the Main Theorem. The only difference is that now the roles of the left endpoint li and the right endpoint ri are interchanged. The reason for this switch is that by playing with the tail of γi we can decrease r(γi) by a small amount, and similarly we can increase Φ(γi ) by a small controlled amount. Case 1 of the proof is as trivial here as in the proof of the Main Theorem. Case 2 is actually simpler here, and can be done with elementary means, because we no longer need to apply Theorem 2.11, and the small growth of Φ is controlled here by the lemmas from §3.
NON-COMPUTABLE JULIA SETS
27
By keeping the same bounds on the variation one finalizes the argument exactly as in the proof of the Main Theorem. 5. Concluding remarks Let us outline here how the methods of [BC1] can be applied to prove Theorem 4.1 instead of the estimates of §3 (we note that a newer version of the same result exists [ABC], where the arguments we quote are simplified). The main technical result of that paper is the following. Let α = [a0 , a1 , . . .] be a Brjuno number, and as before denote pk /qk the sequence of its continued fraction approximants. Let A > 1 and for each integer n ≥ 0 set α[n] = [a0 , a1 , . . . , an , Aqn , 1, 1, 1, . . .]. Then for this particular sequence of Brjuno approximants of α, Φ(α[n]) −→ Φ(α) + log A, n→∞
and moreover, lim r(α[n]) = r(α)/A. The last equality can be used to construct the “drops” in the value of the conformal radius of the Siegel disk needed to inductively fool the TM’s. In this way, one obtains a sequence of Brjuno numbers θi → θ with conformal radii ri = r(θi ) > ri+1 such that lim ri = r > 0, and θi is not computable by all TM’s up to i-th. It remains to show that r(θ) = r, as a priori only the inequality “≤” is known. Buff and Ch´eritat demonstrate it in their context. The idea is roughly speaking in showing that the boundary of ∆(θi ) is well approximated by a periodic cycle of a high period. The perturbation θi 7→ θi+1 is then chosen sufficiently small so that the cycle does not move much. As a final remark, let us point out: Remark 5.1. Combining the methods of [BC1] with our argument as outlined here one may strengthen Theorem 4.1 by showing that there exists a non-computable Siegel Julia set for which the boundary of the Siegel disk is smooth.
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References [BCSS] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, Springer, New York, 1998. [ABC] A. Avila, X. Buff, A. Ch´eritat. Siegel disks with smooth boundaries. Preprint. [BC1] X. Buff, A. Ch´eritat, Quadratic Siegel disks with smooth boudaries. Preprint Univ. Paul Sabatier, Toulouse, III, Num. 242. [BC2] X. Buff, A. Ch´eritat, The Yoccoz Function Continuously Estimates the Size of Siegel Disks, Preprint, 2003. [Bru] A.D. Brjuno. Analytic forms of differential equations, Trans. Mosc. Math. Soc 25(1971) [Brv] M. Braverman, “Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are PolyTime Computable”, Thesis, University of Toronto, 2004, and Proc. CCA 2004, to appear. [Do] A. Douady. Does a Julia set depend continuously on the polynomial? In Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets. ed. R.L. Devaney, Proc. of Symposia in Applied Math., Vol 49, Amer. Math. Soc., 1994, pp. 91-138. [DH1] A. Douady, J.H. Hubbard. Etude dynamique des polynˆomes complexes, I-II. Pub. Math. d’Orsay, 1984. ´ Norm. [DH2] A. Douady, J.H. Hubbard. On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Sup., 18(1985), 287-343. [Ko1] K. Ko, Complexity Theory of Real Functions, Birkh¨auser, Boston, 1991. [Ko2] K. Ko, Polynomial-time computability in analysis, in ”Handbook of Recursive Mathematics”, Volume 2 (1998), Recursive Algebra, Analysis and Combinatorics, Yu. L. Ershov et al. (Editors), pp 1271-1317. [MMY] S. Marmi, P. Moussa, J.-C. Yoccoz, The Brjuno functions and their regularity properties, Commun. Math. Phys. 186(1997), 265-293. [Mat] Y. Matiyasevich, Hilbert’s Tenth Problem, The MIT Press, Cambridge, London, 1993. [McM1] C. McMullen. Complex dynamics and renormalization. Annals of Math. Studies, v.135, Princeton Univ. Press, 1994. [Mil] J. Milnor. Dynamics in one complex variable. Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999. [Pom] C. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, 1992. [RW] R. Rettinger, K. Weihrauch, The Computational Complexity of Some Julia Sets, in STOC’03, June 9-11, 2003, San Diego, California, USA. [Ret] R. Retinger. A fast algorithm for Julia sets of hyperbolic rational functions, Proc. of CCA 2004, to appear. [Sie] C. Siegel, Iteration of analytic functions. Ann. of Math. (2) 43, (1942). 607–612 [Sip] M. Sipser, Introduction to the Theory of Computation, PWS Publishing Company, 1997. [Tur] A. M. Turing, On Computable Numbers, With an Application to the Entscheidungsproblem. In Proceedings, London Mathematical Society, 1936, pp. 230-265. [Wei] K. Weihrauch, Computable Analysis, Springer, Berlin, 2000. [Yoc] J.-C. Yoccoz, Petits diviseurs en dimension 1, S.M.F., Ast´erisque, 231(1995).
NON-COMPUTABLE JULIA SETS M. BRAVERMAN, M. YAMPOLSKY Abstract. We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
1. Summary of the paper Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most na¨ıve work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor’s distance estimator algorithm [Mil], uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show: Main Theorem There exists a parameter value c ∈ C such that the Julia set of the quadratic polynomial fc (z) = z 2 + c is not computable. The structure of the paper is as follows. In the Introduction we discuss the question of computability of real sets, and make the relevant definitions. Further in this section we briefly introduce the reader to the main concepts of Complex Dynamics, and discuss the properties of Julia sets relevant to us. In the end of the Introduction, we outline the conceptual idea of the proof of Main Theorem. Section §3 contains the technical lemmas on which the argument is based. In §4 we complete the proof. Acknowledgement. We wish to thank Arnaud Ch´eritat for helpful comments on an earlier draft of this paper. The first author would like to thank Stephen Cook for many discussions on computability of real sets. Date: March 6, 2008. The first author’s research is supported by an NSERC CGS scholarship. The second author’s research is supported by NSERC operating grant. 1
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M. BRAVERMAN, M. YAMPOLSKY
2. Introduction 2.1. Introduction to the Computability of Real Sets. Classical Computability. The computability theory in general allows us to classify problems into the tractable (“computable”) and intractable (“uncomputable”). All the common computational tasks such as integer operations, list sorting, etc. are easily seen to be computable. On the other hand, there are many uncomputable problems. In the formal setting for the study of computability theory computations are performed by objects called the Turing Machines. Turing Machines were introduced in 1936 by Alan Turing (see [Tur]) and is accepted by the scientific community as the standard model of computation. The Turing Machine (TM in short) is capable of solving exactly the same problems as an ordinary computer. Most of the time, one can think of the TM as a computer program written in any programming language. It is important to mention that there are only countably many TMs, which can be enumerated in a natural way. See [Sip] for a formal discussion on TMs. We define computability as follows. Definition 2.1. We say that a function f : {0, 1}∗ → {0, 1}∗ is computable if there is a TM M, which on input string s outputs the string f (s). We say that the set L ⊂ {0, 1}∗ is computable or decidable if its characteristic function χL : {0, 1}∗ → {0, 1} is computable. While most “common” functions are computable, there are uncountably many uncomputable functions and undecidable sets. The most famous intractable problems are the Halting Problem and the solvability of a Diophantine equation (Hilbert’s 10-th problem), see [Sip] and [Mat] for more information.
Computability of Real Functions and Sets. In the present paper we are interested in the computability of functions f : Rn → R and subsets of Rn , particularly subsets of R2 ∼ = C. We cannot directly apply definition 2.1 here, since real number cannot be represented in general by finite sequences of bits. Denote by D the set of the dyadic rationals, i.e. rationals of the form 2pm . Rationals in D can be easily represented as binary numbers. We say that φ : N → D is an oracle for a real number x, if |x − φ(n)| < 2−n for all n ∈ N. In other words, φ provides a good dyadic approximation for x. We say that a TM M φ is an oracle machine, if at any step of the computation M is allowed to query the value φ(n) for any n. This definition allows us to define the computability of real functions on compact sets. Definition 2.2. We say that a function f : [a, b] → [c, d] is computable, if there exits an oracle TM M φ (m) such that if φ is an oracle for x ∈ [a, b], then on input m, M φ outputs a y ∈ D such that |y − f (x)| < 2−m . In other words, with an access to arbitrarily good approximations for x, M should be able to produce an arbitrarily good approximation for f (x). This definition trivially generalizes to domains of higher dimension. See [Ko1] for more details. One of the most important properties of computable functions is that
NON-COMPUTABLE JULIA SETS
3
Proposition 2.1. Computable functions are continuous. Let K ⊂ Rk be a compact set. We would like to give a definition for K being computable. Note that saying by analogy to definition 2.1, that C is computable if and only if the characteristic function χC is computable does not work here. Recall that only continuous functions can be computable, hence χK is not computable unless K = ∅. We say that a TM M computes the set K if it approximates K in the Hausdorff metric. Recall that the Hausdorff metric is a metric on compact subsets of Rn defined by (2.1)
dH (X, Y ) = inf{ǫ > 0 | X ⊂ Uǫ (Y ) and Y ⊂ Uǫ (X)},
where Uǫ (S) is defined as the union of the set of ǫ-balls with centers in S. We approximate C using a class C of sets which is dense in metric dH among the compact sets and which has a natural correspondence to binary strings. Namely C is the set of finite unions of dyadic balls: (n ) [ C= B(di , ri ) | where di , ri ∈ D . i=1
Members of C can be encoded as binary strings in a natural way. The following definition is equivalent to the set computability definition given in [Wei] (see also [RW]).
Definition 2.3. We say that a compact set K ⊂ Rk is computable, if exists a TM M(m), such that on input m, M(m) outputs an encoding of Cm ∈ C such that dH (K, Cm ) < 2−m .
To illustrate the robustness of this definition we present the following two equivalent characterizations of computable sets. The first one relates the definition to computer graphics. It is not stated precisely here, but it can be easily made precise. The second one relates the computability of sets to the computability of functions as per definition 2.2. Theorem 2.2. For a compact K ⊂ Rk the following are equivalent: (1) K is computable as per definition 2.3, (2) (in the case k = 1, 2) K can be drawn on a computer screen with arbitrarily good precision, (3) the distance function dK (x) = inf{|x − y| | y ∈ K} is computable as per definition 2.2. In the present paper we are interested in questions concerning the computability of the Julia set Jc = J(fc ) = J(z 2 + c) (see the next section for the definition). Since there are uncounably many possible parameter values for c, we cannot expect for each c to have a machine M such that M computes Jc (recall that there are countably many TMs). On the other hand, it is reasonable to want M to compute Jc with an oracle access to c. Define the function J : C → K ∗ (K ∗ is the set of all compact subsets of C) by J(c) = J(fc ). In a complete analogy to definition 2.2 we can define Definition 2.4. We say that a function f : S → K ∗ for some bounded set S is computable, if there exits an oracle TM M φ (m) such that if φ is an oracle for x ∈ S, then on input m, M φ outputs a C ∈ C such that dH (C, f (x)) < 2−m .
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M. BRAVERMAN, M. YAMPOLSKY
In the case of Julia sets: Definition 2.5. We say that Jc is computable if the function J : d 7→ Jd is computable on the set {c}. The following has been shown (see [Brv], [Ret]): Theorem 2.3. Denote by H the set of parameters c for which Jc is hyperbolic, then (i) Jc is computable for all c ∈ H, moreover (ii) the function J is computable on B(0, b) ∩ H for all b > 0. Our goal in this paper is to show that there are values of c for which Jc is not computable under definition 2.5, which is the weakest possible definition in this setting. We will be using the following version of theorem 2.1 for set functions. Theorem 2.4. Suppose that a TM M φ computes the function J on a set S, then J is continuous on S in Hausdorff metric. In the next section we proceed to define Julia sets of rational maps and review their basic properties. In particular, towards the end of the introduction, we will see a mechanism by which the continuity required by Theorem 2.4 may fail. It should be noted that the question of computability of dynamically generated fractal sets, such as Julia sets, has been discussed by Blum, Cucker, Shub, and Smale in [BCSS]. The definition of set computability used in [BCSS] is, however, quite different from definition 2.3. The BCSS model allows precise arithmetic, but requires completely accurate pictures to be generated. Under this definition all Julia sets but the most trivial ones can be shown to be uncomputable. Our definition, however, reflects the hardness of generating pictures of the set, and the fact that we have good images of Julia sets in some cases shows the usefulness of this definition in reflecting the true “hardness” of the set. 2.2. Julia sets of polynomial mappings. We recall the main definitions of complex dynamics relevant to our result only briefly; a good general reference is the book of Milnor [Mil]. For a rational mapping R of degree deg R = d ≥ 2 considered as a dynamical system on the Riemann sphere ˆ →C ˆ R:C the Julia set is defined as the complement of the set where the dynamics is Lyapunov-stable: ˆ having an open neighborhood U(z) Definition 2.6. Denote F (R) the set of points z ∈ C n on which the family of iterates R |U (z) is equicontinuous. The set F (R) is called the Fatou ˆ \ F (R) is the Julia set. set of R and its complement J(R) = C In the case when the rational mapping is a polynomial P (z) = a0 + a1 z + · · · + ad z d : C → C
an equivalent way of defining the Julia set is as follows. Obviously, there exists a neighˆ on which the iterates of P uniformly converge to ∞. Denoting A(∞) borhood of ∞ on C
NON-COMPUTABLE JULIA SETS
5
the maximal such domain of attraction of ∞ we have A(∞) ⊂ F (R). We then have J(P ) = ∂A(∞). ˆ \ cl A(∞) is called the filled Julia set, and denoted K(P ); it consists of The bounded set C points whose orbits under P remain bounded: ˆ sup |P n (z)| < ∞}. K(P ) = {z ∈ C| n
For future reference, let us list in a proposition below the main properties of Julia sets: ˆ →C ˆ be a rational function. Then the following properties Proposition 2.5. Let R : C hold: ˆ which is completely invariant: R−1 (J(R)) = • J(R) is a non-empty compact subset of C J(R); • J(R) = J(Rn ) for all n ∈ N; • J(R) is perfect; ˆ • if J(R) has non-empty interior, then it is the whole of C; ˆ be any open set with U ∩ J(R) 6= ∅. Then there exists n ∈ N such that • let U ⊂ C n R (U) ⊃ J(R); • periodic orbits of R are dense in J(R). Let us further comment on the last property. For a periodic point z0 = Rp (z0 ) of period p its multiplier is the quantity λ = λ(z0 ) = DRp (z0 ). We may speak of the multiplier of a periodic cycle, as it is the same for all points in the cycle by the Chain Rule. In the case when |λ| = 6 1, the dynamics in a sufficiently small neighborhood of the cycle is governed by the Intermediate Value Theorem: when |λ| < 1, the cycle is attracting (super-attracting if λ = 0), if |λ| > 1 it is repelling. Both in the attracting and repelling cases, the dynamics can be locally linearized: (2.2)
ψ(Rp (z)) = λ · ψ(z)
where ψ is a conformal mapping of a small neighborhood of z0 to a disk around 0. By a classical result of Fatou, a rational mapping has at most finitely many non-repelling periodic orbits. Therefore, we may refine the last statement of Proposition 2.5: • repelling periodic orbits are dense in J(R). In the case when λ = e2πiθ , θ ∈ R, the simplest to study is the parabolic case when θ = n/m ∈ Q, so λ is a root of unity. In this case Rp is not locally linearizable; it is not hard to see that z0 ∈ J(R). In the complementary situation, two non-vacuous possibilities are considered: Cremer case, when Rp is not linearizable, and Siegel case, when it is. In the latter case, the linearizing map ψ from (2.2) conjugates the dynamics of Rp on a neighborhood U(z0 ) to the irrational rotation by angle θ (the rotation angle) on a disk around the origin. The maximal such neighborhood of z0 is called a Siegel disk. Siegel disks will prove crucial to our study, and will be discussed in more detail in the next section.
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M. BRAVERMAN, M. YAMPOLSKY
To conclude the discussion of the basic properties of Julia sets, let us consider the simplest exampes of non-linear rational endomorphisms of the Riemann sphere, the quadratic polynomials. Every affine conjugacy class of quadratic polynomials has a unique representative of the form fc (z) = z 2 + c, the family fc (z) = z 2 + c, c ∈ C
is often referred to as the quadratic family. For a quadratic map the structure of the Julia set is governed by the behaviour of the orbit of the only finite critical point 0. In particular, the following dichotomy holds: Proposition 2.6. Let K = K(fc ) denote the filled Julia set of fc , and J = J(fc ) = ∂K. Then: • 0 ∈ K implies that K is a connected, compact subset of the plane with connected complement; • 0∈ / K implies that K = J is a planar Cantor set. The Mandelbrot set M ⊂ C is defined as the set of parameter values c for which J(fc ) is connected. Continuity of the dependence c 7→ J(fc ). A natural question to pose for polynomials in the quadratic family is whether the Julia set varies continuously with the parameter c. To make sense of this question, recall the definition of the Hausdorff distance distH between compact sets X, Y in the plane (2.1). It turns out that the dependence c 7→ J(fc ) is discontinuous in the Hausdorff distance. For an excellent survey of this problem see the paper of Douady [Do]. The discontinuity which has found most interesting dynamical applications occurs at parameter values for which fc has a parabolic point. We, however, will employ a more obvious discontinuity which is related to Siegel disks. Let us first note that by a result of Douady and Hubbard [DH1] a quadratic polynomial has at most one non-repelling cycle in C. In particular, there is at most one cycle of Siegel disks. Proposition 2.7. Let c∗ ∈ M be a parameter value for which fc has a Siegel disk. Then the map c 7→ J(fc ) is discontinuous at c∗ . Proof. Let z0 be a Siegel periodic point of fc and denote ∆ the Siegel disk around ζ0 , p its period, and θ the rotation angle. By the Implicit Function Theorem, there exists a holomorphic mapping ζ : U(c∗ ) → C such that ζ(c∗) = z0 and ζ(c) is fixed under (fc )p . The mapping ν : c 7→ D(fc )p (ζ(c)) is holomorphic, hence it is either constant or open. If it is constant, all quadratic polynomials have a Siegel disk. This is not possible: for instance, f1/4 has a parabolic fixed point, and thus no other non-repelling cycles. Therefore, ν is open, and in particular, there is a sequence of parameters cn → c∗ such that ζ(cn ) has multiplier e2πipn /qn . Since ζ(cn ) is parabolic, it lies in the Julia set of fcn . Hence distH (J(fc∗ ), J(fcn )) > dist(c∗ , ∂∆)/2
NON-COMPUTABLE JULIA SETS
7
for n large enough.
Thus an arbitrarily small change of the multiplier of the Siegel point may lead to an implosion of the Siegel disk – its inner radius collapses to zero. Siegel disks of quadratic maps. Let us discuss in more detail the occurence of Siegel disks in the quadratic family. For a number θ ∈ [0, 1) denote [r0 , r1 , . . . , rn , . . .], ri ∈ N ∪ {∞} its possibly finite continued fraction expansion: (2.3)
[r0 , r1 , . . . , rn , . . .] ≡
1
1
r0 +
1
r1 +
1 rn + · · · Such an expansion is defined uniqely if and only if θ ∈ / Q. In this case, the rational convergents pn /qn = [r0 , . . . , rn−1 ] are the closest rational approximants of θ among the numbers with denominators not exceeding qn . In fact, setting λ = e2πiθ , we have ···+
|λh − 1| > |λqn − 1| for all 0 < h < qn+1 , h 6= qn .
The difference |λqn − 1| lies between 2/qn+1 and 2π/qn+1 , therefore the rate of growth of the denominators qn describes how well θ may be approximated with rationals. Definition 2.7. The diophantine numbers of order k, denoted D(k) is the following class of irrationals “badly” approximated by rationals. By definition, θ ∈ D(k) if there exists c > 0 such that qn+1 < cqnk−1 The numbers qn can be calculated from the recurrent relation qn+1 = rn qn + qn−1 , with q0 = 0, q1 = 1. Therefore, θ ∈ D(2) if and only if the sequence {ri } is bounded. Dynamicists call such numbers bounded type (number-theorists prefer constant type). An extreme example of a number of bounded type is the golden mean √ 5−1 θ∗ = = [1, 1, 1, . . .]. 2 The set \ Dk D(2+) ≡ k>2
has full measure in the interval [0, 1). In 1942 Siegel showed:
ˆ of period p. Theorem 2.8 ([Sie]). Let R be an analytic map with an periodic point z0 ∈ C Suppose the multiplier of the cycle λ = e2πiθ with θ ∈ D(2+),
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M. BRAVERMAN, M. YAMPOLSKY
then the local linearization equation (2.2) holds. The strongest known generalization of this result was proved by Brjuno in 1972: Theorem 2.9 ([Bru]). Suppose (2.4)
B(θ) =
X log(qn+1 ) n
qn
< ∞,
then the conclusion of Siegel’s Theorem holds.
Note that a quadratic polynomial with a fixed Sigel disk with rotation angle θ after an affine change of coordinates can be written as (2.5)
Pθ (z) = z 2 + e2πiθ z.
In 1987 Yoccoz [Yoc] proved the following converse to Brjuno’s Theorem: Theorem 2.10 ([Yoc]). Suppose that for θ ∈ [0, 1) the polynomial Pθ has a Siegel point at the origin. Then B(θ) < ∞. The numbers satisfying (2.4) are called Brjuno numbers; the set of all Brjuno numbers will be denoted B. It is evident that ∪D(k) ⊂ B. The sum of the series (2.4) is called the Brjuno function. For us a different characterization of B will be more useful. Inductively define θ1 = θ and θn+1 = {1/θn }. In this way, θn = [rn−1 , rn , rn+1, . . .].
We define the Yoccoz’s Brjuno function as ∞ X 1 Φ(θ) = θ1 θ2 · · · θn−1 log . θn n=1
One can verify that
B(θ) < ∞ ⇔ Φ(θ) < ∞. The value of the function Φ is related to the size of the Siegel disk in the following way. Definition 2.8. Let P (θ) be a quadratic polynomial with a Siegel disk ∆θ ∋ 0. Consider the conformal isomorphism φ : D 7→ ∆ fixing 0. The conformal radius of the Siegel disk ∆θ is the quantity r(θ) = |φ′ (0)|. For all other θ ∈ [0, ∞) we set r(θ) = 0. By the Koebe One-Quarter Theorem of classical complex analysis, the internal radius of ∆θ is at least r(θ)/4. Yoccoz [Yoc] has shown that the sum Φ(θ) + log r(θ) is bounded below independently of θ ∈ B. Recently, Buff and Ch´eritat have greatly improved this result by showing that:
NON-COMPUTABLE JULIA SETS
9
Theorem 2.11 ([BC2]). The function θ 7→ Φ(θ) + log r(θ) extends to R as a 1-periodic continuous function. We remark that the following stronger conjecture exists (see [MMY]): Marmi-Moussa-Yoccoz Conjecture. [MMY] The function θ 7→ Φ(θ)+log r(θ) is H¨older of exponent 1/2. Dependence of the conformal radius of a Siegel disk on the paramater. In this section we will show that the conformal radius of a Siegel disk varies continuously with the Julia set. To that end we will need a preliminary definition: Definition 2.9. Let (Un , un ) be a sequence of topological disks Un ⊂ C with marked points un ∈ Un . The kernel or Carath´eodory convergence (Un , un ) → (U, u) means the following: • un → u; • for any compact K ⊂ U and for all n sufficiently large, K ⊂ Un ; • for any open connected set W ∋ u, if W ⊂ Un for infinitely many n, then W ⊂ U. The topology on the set of pointed domains which corresponds to the above definition of convergence is again called kernel or Carath´eodory topology. The meaning of this topology is as follows. For a pointed domain (U, u) denote φ(U,u) : D → U
the unique conformal isomorphism with φ(U,u) (0) = u, and (φ(U,u) )′ (0) > 0. By the Riemann Mapping Theorem, the correspondence ι : (U, u) 7→ φ(U,u) establishes a bijection between marked topological disks properly contained in C and univalent maps φ : D → C with φ′ (0) > 0. The following theorem is due to Carath´eodory, a proof may be found in [Pom]: Theorem 2.12 (Carath´ eodory Kernel Theorem). The mapping ι is a homeomorphism with respect to the Carath´eodory topology on domains and the compact-open topology on maps. Proposition 2.13. The conformal radius of a quadratic Siegel disk varies continuously with respect to the Hausdorff distance on Julia sets. Proof. To fix the ideas, consider the family Pθ with θ ∈ B and denote ∆θ the Siegel disk of Pθ . It is easy to see that the Hausdorff convergence J(Pθn ) → J(Pθ ) implies the Carath´eodory convergence of the pointed domains (∆θn , 0) → (∆, 0). The proposition follows from this and the Carath´eodory Kernel Theorem.
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M. BRAVERMAN, M. YAMPOLSKY
Non-computability of the Yoccoz’s Brjuno function. By the analysis similar to our proof of the Main Theorem, we can also prove a direct non-computability result for the Yoccoz’s Brjuno function Φ: Theorem 2.14 (Non-computability of Φ). There exists a parameter value θ ∈ R/Z such that Φ(θ) is not computable by any oracle Turing Machine with access to θ. It is worth noting that Marmi-Moussa-Yoccoz Conjecture as stated above and Theorem 2.14 imply the existence of a non-computable quadratic Julia set. To see this, we first formulate: Conditional Implication . If Marmi-Moussa-Yoccoz Conjecture holds, then the function ν : θ 7→ Φ(θ) + log r(θ) is computable by one Turing Machine on the entire interval [0, 1]. We use the following result of Buff and Ch´eritat ([BC2]). Lemma 2.15 ([BC2]). For any rational point θ =
p q
∈ [0, 1] denote, as before,
Pθ (z) = e2πiθ z + z 2 , and let the Taylor expansion of Pθ◦q (z) at 0 start with Pθ◦q (z) = z + Az q+1 + . . . , for q ∈ N 1/q 1 . Denote by Φtrunc the modification of Φ applied to rational numbers Let L(θ) = qA where the sum is truncated before the infinite term. Then we have the following explicit formula for computing ν(θ): (2.6)
ν(θ) = Φtrunc (θ) + log L(θ) +
log 2π . q
Equation (2.6) allows us to compute the value of ν easily at every rational θ ∈ Q∩[0, 1] with an arbitrarily good precision. In addition, assuming the conjecture, we have |ν(x)−ν(y)| < 2−n whenever |x − y| < c · 2−2n for some constant c, hence ν has an (easily) computable modulus of continuity. These two facts together imply that ν is computable by a single machine of the interval [0, 1] (see for example Proposition 2.6 in [Ko2]). This implies the Conditional Implication. The following conditional result follows: Lemma 2.16 (Conditional). Suppose the Conditional Implication holds. Let θ ∈ [0, 1] be such that Φ(θ) is finite. Then there is an oracle Turing Machine M1φ computing Φ(θ) with an oracle access to θ if and only if there is an oracle Turing Machine M2φ computing r(θ) with an oracle access to θ.
NON-COMPUTABLE JULIA SETS
11
Proof. Suppose that M1φ computes Φ(θ) for some θ. Let M φ be the machine uniformly computing the function ν. Then we can use M1φ and M φ to compute log r(θ) = ν(θ) − Φ(θ) with an arbitrarily good precision. We can then use this construction to give a machine M2φ which computes r(θ). The opposite direction is proved analogously. Lemma 2.16 with Theorem 2.14 imply that there is a θ for which r(θ) is non-computable. Considerations of the Riemann Mapping imply that for this value of θ the Julia set of Pθ is also non-computable. Note that for the proof of Conditional Implication we did not need the full power of the conjecture. All we needed is some computable bound on the modulus of continuity of ν. Outline of the construction of a non-computable quadratic Julia set. We are now prepared to outline the idea of our construction. The outline given below is rather rough and suffers from obvious logical deficiencies. However, it captures the idea of the proof in a simple to understand form. Suppose that every Julia set of a polynomial Pθ is computable by an oracle machine M φ , where φ represents θ. There are countably many machines, so we can enumerate them M1φ , M2φ , . . .. Denote by Si the domain on which Miφ computes JPθ properly. Then we must have: S (1) C = ∞ i=1 Si , (2) for each i, the function J : θ 7→ J(Pθ ) is continuous on Si .
Figure 1. The Siegel disks of Pθ for θ given by the continued fractions [1, 1, 1, . . .], [1, 1, 1, 20, 1, . . .], and [1, 1, 1, 20, 1, 1, 1, 30, 1, . . .] Let us start with a machine Mnφ1 which computes J(Pθ∗ ) for θ∗ = [1, 1, 1, . . .]. If any of the digits ri in this infinite continued fraction is changed to a sufficiently large N ∈ N,
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M. BRAVERMAN, M. YAMPOLSKY
the conformal radius of the Siegel disk will become small. For N → ∞ the Siegel disk will implode and its center will become a parabolic fixed point in the Julia set. Given the continuity of the dependence of the conformal radius of the Siegel disk on the Julia set, we have the following: There exists i1 > 1 such that for every θ1 whose continued fraction starts with i1 ones, for the Julia set of Pθ1 to be computable by Mnφ1 , it must possess a Siegel disk of a conformal radius r(θ1 ) > r(θ∗ )(1 − 1/8). We can thus “fool” the machine Mnφ1 by selecting θ1 given by a continued fraction where all digits are ones except ri1 = N1 >> 1. If we are careful, we can do it so that (2.7)
r(θ∗ )(1 − 1/4) < r(θ1 ) < r(θ∗ )(1 − 1/8).
To “fool” the machine Mnφ2 we then change a digit ri2 for i2 > i1 sufficiently far in the continued fraction of θ1 to a large N2 . In this way, we will obtain a Brjuno number θ2 for which (2.8)
r(θ∗ )(1 − 1/4 − 1/8) < r(θ2 ) < r(θ∗ )(1 − 1/4).
Continuing in this manner we will arrive at a limiting Brjuno number θ∞ for which the Julia set is uncomputable. To make such a scheme work, we need a careful analysis of the dependence of the conformal radius on the parameter. In this a key role is played by Theorem 2.11 of Buff and Ch´eritat which allows us to obtain a controlled change in the value of r(α) by changing Φ(α). The relevant analysis is carried out in the next section. A note on the connection with [BC1]. A. Ch´eritat has pointed to us that methods of [BC1], where Siegel disks with smooth boundaries are constructed for the quadratic family can be used to derive the Main Theorem. We discuss this in the section following the proofs of the main theorems. We note here that the argument we give is based on quite elementary estimates of the function Φ and is thus accessible to non-dynamicists. It has an added advantage of yielding Theorem 2.14. 3. Making Small Changes to Φ 3.1. Small Changes to Φ. A key step of the construction outlined above is making careful adjustments of r(θi ) as in the first two steps (2.7) and (2.8) above. We do not have a direct control over the value of r(α), but Buff and Ch´eritat’s Theorem 2.11 shows that small decreases of r(α) we would like to make correspond to a small controlled increment of the value of Φ(α). Estimates of a similar nature has appeared in the works of various authors (compare, for example, with [BC1]). For a number γ = [a1 , a2 , . . .] ∈ R \ Q we denote 1 αi (γ) = , 1 ai + 1 ai+1 + ai+2 + . . .
NON-COMPUTABLE JULIA SETS
so that Φ(γ) =
X
α1 (γ)α2 (γ) . . . αn−1 (γ) log
n≥1
13
1 . αn (γ)
The main goal of this section is to prove the following lemma: Lemma 3.1. For any initial segment I = (a0 , a1 , . . . , an ), write α = [a0 , a1 , . . . , an , 1, 1, 1, . . . ]. Then for any ε > 0, there is an m > 0 and an integer N such that if we write β = [a0 , a1 , . . . , an , 1, 1, . . . , 1, N, 1, 1, . . .], where the N is located in the n + m-th position, and Φ(α) + ε < Φ(β) < Φ(α) + 2ε. The proof is technical and will require some preparation. The idea is to choose an m large enough, so that changing an+m (which will eventually be N) by 1 changes the value of Φ by a very small amount (< ε). When N → ∞, Φ(α) → ∞, hence the value of Φ must hit the interval (Φ(α) + ε, Φ(α) + 2ε). Denote 1 Φ− (α) = Φ(α) − α0 α1 . . . αn+m−1 log . αm+n The value of the integer m > 0 is yet to be determined. Denote β N = [a0 , a1 , . . . , an , 1, 1, . . . , 1, N, 1, 1, . . .]. We prove the following Lemma 3.2. For any N and i ≤ n + m we have N log αi (β ) < 2i−(n+m) . N +1 αi (β )
Proof. We proove the lemma by induction on i, starting from the base case i = n + m, and proceeding down to i = 0. The base case is i = n + m, we want to prove N α (β ) n+m < 1. log αn+m (β N +1 ) We have
rn+m
1 N +x = N +1+x = 1+ 1 , 1 N +x N +x N +1+x < 2, and | log rn+m | < 1.
αn+m (β N ) = = αn+m (β N +1 )
for some x < 1. Hence 1 < rn+m
Induction step. Supposing that the statement is true for i + 1, we prove it for i. We have 1 N ai + αi+1 (β N +1) αi (β ) ai + αi+1 (β N ) = = . 1 αi (β N +1 ) ai + αi+1 (β N ) ai + αi+1 (β N +1 )
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M. BRAVERMAN, M. YAMPOLSKY
Suppose that αi+1 (β N +1 ) ≥ αi+1 (β N ). Then we know that αi+1 (β N +1 ) i+1−(n+m) < e2 , N αi+1 (β )
and we want to prove that ai + αi+1 (β N +1 ) i−(n+m) < e2 , N ai + αi+1 (β ) since this expression is obviously bigger than 1. The situation would have been the same if αi+1 (β N +1 ) ≤ αi+1 (β N ), with the numerator and the denominator exchanged. In other words, it is enough to prove that for 0 < d < c < 1 and a pair of integers r ≥ 0 and k ≥ 1, k+c c −r −r−1 < e2 implies that < e2 . d k+d First of all, it is easy to see that for k ≥ 1, k+c 1+c ≤ , k+d 1+d hence it suffices to show that −r 1/2 1+c −r−1 < e2 = e2 . 1+d Thus we need to demonstrate that 1 + c c 1/2 . < 1+d d This is equivalent to (1 + 2c + c2 )d < (1 + 2d + d2 )c ⇔ d + c2 d < c + d2 c ⇔ cd(c − d) < c − d.
The last inequality holds, since cd < 1 and c − d > 0.
The following lemma is proven exactly as the previous one with a shifted induction base. Lemma 3.3. For any N and i < n + m we have N α (β ) i log < 2i−(n+m) . αi (β 1 )
Proof. The proof goes by induction exactly as before. We need to verify the base case i = n + m − 1. For this value of i, √ 5−1 1 , αn+m−1 (β 1 ) = = φ 2 and 1 1 < 1. ≤ αn+m−1 (β N ) = 1 φ 1 + N +...
NON-COMPUTABLE JULIA SETS
Hence we have
N α (β ) n+m−1 < log φ ≈ 0.481 < 0.5 = 2−1 . log 1 αn+m−1 (β )
We now bound the influence of the change on the log
15
1 terms. αi
Lemma 3.4. For any N and i < n + m, 1 log N) α (β i < 2i−(n+m)+1 . log 1 log N +1 α (β ) i
N
Proof. Assume that αi (β ) ≤ αi (β case we need to prove
N +1
), the reverse case is done in the same way. In this
1 αi (β N ) i−(n+m)+1 < e2 . 1 log αi (β N +1 ) log
1 and αi (β N +1 ) = Denote c = αi+1 (β N ) and d = αi+1 (β N +1 ). Then we have αi (β N ) = k+c 1 for some integer k ≥ 1. Hence 0 < d ≤ c < 1. We have k+d 1 log log(k + c) αi (β N ) . = 1 log(k + d) log αi (β N +1 ) c log(k + c) c i−(n+m)+1 By lemma 3.2 we know that < e2 , hence it suffices to show that ≤ . d log(k + d) d log(k + d) log(k + x) log(k + c) ≤ . Consider the function f (x) = on This is equivalent to c d x the interval (0, 1). The reader can readily verify that f ′ (x) < 0 for x ∈ (0, 1) so that f is decreasing on this interval, and hence f (c) ≤ f (d), which completes the proof. In the same way that Lemma 3.4 follows from Lemma 3.2, the following Lemma follows from Lemma 3.3 Lemma 3.5. For any N and i < n + m − 1, 1 log N) α (β i log < 2i−(n+m)+1 . 1 log 1 αi (β )
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M. BRAVERMAN, M. YAMPOLSKY
Proof. As in lemma 3.4.
We are now ready to bound the influence of changes in N on the value of Φ− . Lemma 3.6. For any α of the form as in lemma 3.1 and for any ε > 0, there is an m0 > 0 such that for any N and any m ≥ m0 , ε |Φ− (β N ) − Φ− (β 1 )| < . 4 P 1 Proof. The in the expression for Φ(β ) converges, hence there is an m1 > 1 such that the P 1 ε tail of the sum i≥n+m1 α1 α2 . . . αi−1 log < . We will show how to choose m0 ≥ m1 αi 16 to satisfy the conclusion of the lemma. We bound the influence of the change from β 1 to β N using lemmas 3.3 and 3.5. The influence on each of the “head elements” (i < n + m1 ) is bounded by 1 1 1 i−1 α (β ) . . . α (β ) log 1 i−1 1 αi (β ) X j−(n+m) log 2 + 2i−(n+m)+1 < 2i−(n+m)+2 < 2m1 +2−m . < 1 j=1 α1 (β N ) . . . αi−1 (β N ) log αi (β N )
By making m sufficiently large (i.e. by choosing a sufficiently large m0 ) we can ensure that 1 α1 (β N ) . . . αi−1 (β N ) log ε ε αi (β N ) 1− n + m are not affected by the change, and the change decreases αn+m , so that αn+m (β N ) ≤ αn+m (β 1 ). Hence 1 α1 (β N ) . . . αi−1 (β N ) log α1 (β N ) . . . αn+m (β N ) αi (β N ) = log ≤ log 1 α1 (β 1 ) . . . αn+m (β 1 ) α1 (β 1 ) . . . αi−1 (β 1 ) log αi (β 1 ) log
n+m−1 X α1 (β N ) . . . αn+m−1 (β N ) < 2j−(n+m) < 1 α1 (β 1 ) . . . αn+m−1 (β 1 ) j=1
So in this case each term could increase by a factor of e at most. We see that after the change each term of the tail could increase by a factor of e at most. eε The value of the tail remains positive in the interval (0, ], hence the change in the tail 16 3ε eε < . is bounded by 16 16 So the total change in Φ− is bounded by ε 3ε ε change in the “head” + change in the “tail” < + = . 16 16 4 The following Lemma follows immediately from Lemma 3.6.
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Lemma 3.7. For any ε and for the same m0 (ε) as in lemma 3.6, for any m ≥ m0 and N, ε |Φ− (β N ) − Φ− (β N +1 )| < . 2 Proof. We have ε ε ε |Φ− (β N ) − Φ− (β N +1 )| ≤ |Φ− (β N ) − Φ− (β 1 )| + |Φ− (β 1) − Φ− (β N +1 )| < + = . 4 4 2 1 We will now have to take a closer look at the term α1 . . . αn+m−1 log = Φ(α)−Φ− (α). αm+m For this we will need a slightly sharper version of Lemma 3.2. Lemma 3.8. For any N and i ≤ n + m we have N 2i−(n+m) α (β ) i < log . αi (β N +1 ) N
Proof. The induction works exactly as in Lemma 3.2, we only need to verify the base case i = n + m. We have 1 N αn+m (β ) N +1+φ 1 N +φ = = =1+ , N +1 1 αn+m (β ) N +φ N +φ N +1+φ hence 1< So
1 αn+m (β N ) 1, αk−1 αk < . 2 Proof. There is an integer l ≥ 1 such that 1 1 1 αk < αk = . αk−1 αk = l + αk αk + αk 2 Denote Φ1 (α) = α1 . . . αn+m−1 log following.
1 αn+m
= Φ(α) − Φ− (α), we are now ready to prove the
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19
Lemma 3.10. For sufficiently large m, for any N, ε Φ1 (β N +1 ) − Φ1 (β N ) < . 2 Proof. According to Lemma 3.8 we have n+m−1 N +1 N +1 X α (β ) . . . α (β ) 1 1 n+m−1 i−(n+m) log < 2 /N < . α1 (β N ) . . . αn+m−1 (β N ) N i=1
Hence α1 (β N +1 ) . . . αn+m−1 (β N +1 ) < α1 (β N ) . . . αn+m−1 (β N )e1/N , and 1 log log(N + 1 + φ) αn+m (β N +1 ) = Φ1 (β N )e1/N . Φ1 (β N +1 ) < Φ1 (β N )e1/N 1 log(N + φ) log αn+m (β N )
Hence
1/N log(N + 1 + 1/φ) −1 < Φ (β ) − Φ (β ) < Φ (β ) e log(N + 1/φ) e log(N + 1 + 1/φ) 1 N Φ (β ) (1 + ) −1 . N log(N + 1/φ) log(N + 1 + 1/φ) , then (N + 1/φ)x = We make the following calcualtions. Denote x = log(N + 1/φ) N + 1 + 1/φ, and 1 1 < e N + 1/φ . (N + 1/φ)x−1 = 1 + N + 1/φ 3 3 3 N + 1/φ > e1/3 , and so x − 1 < < , thus x < 1 + . N + 1/φ N N 1
N +1
1
N
1
N
By Lemma 3.9 we have 1 Φ (β ) = α1 (β ) . . . αn+m−1 (β ) log < αn+m (β N ) 1
N
N
N
Thus
(n+m−1)/2 1 log(N + 1/φ). 2
e log(N + 1 + 1/φ) Φ (β ) − Φ (β ) < Φ (β ) (1 + ) −1 < N log(N + 1/φ) (n+m−1)/2 (n+m−1)/2 14 1 1 log(N + 1/φ) ((1 + e/N)(1 + 3/N) − 1) < log(N + 1/φ) . 2 2 N 14 ε Since ∈ o(1/ log(N + 1/φ)), this expression can be always made less than by choosing N 2 m large enough. 1
N +1
1
N
1
N
Lemmas 3.7 and 3.10 yield the following
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M. BRAVERMAN, M. YAMPOLSKY
Lemma 3.11. For sufficiently large m, for any N, Φ(β N +1 ) − Φ(β N ) < ε. Proof. We use Lemmas 3.7 and 3.10. For sufficiently large m, Φ(β N +1 ) − Φ(β N ) ≤ Φ− (β N +1 ) − Φ− (β N ) + Φ1 (β N +1 ) − Φ1 (β N )
and
1 · α1 (β 1 ) . . . αn+m−1 (β 1 ) e
1 log(N + 1/φ) 1 1 · Φ (β ). e log(1 + 1/φ) The latter expression obviously goes to ∞ as N → ∞. Φ1 (β N ) >
We are now ready to prove Lemma 3.1.
Proof. (of Lemma 3.1). Choose m large enough for lemma 3.11 to hold. Increase N by one at a time starting with N = 1. We know that Φ(β 1 ) = Φ(α) < Φ(α) + ε, and by Lemma 3.12, there exists an M with Φ(β M ) > Φ(α) + ε. Let N be the smallest such M. Then Φ(β N −1 ) ≤ Φ(α) + ε, and by Lemma 3.11 Φ(β N ) < Φ(β N −1 ) + ε ≤ Φ(α) + 2ε.
Hence Choosing β = β N
Φ(α) + ε < Φ(β N ) < Φ(α) + 2ε. completes the proof.
The following is an important note on the proof which we will use later in the section. Lemma 3.13. After the process described above denote the total variation by X 1 1 N N N 1 1 1 α1 (β )α2 (β ) . . . αi−1 (β ) log . T = − α (β )α (β ) . . . α (β ) log 1 2 i−1 αi (β N ) αi (β 1 ) i≥0
Then T is finite, and satisfies T ≤ 3ε.
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21
Proof. Note that m > m1 . We split T into two parts: the variation T1 on the first n+m1 −1 terms of the sum, and the variation T2 on the rest of the terms (see proof of lemma 3.6 for the definition of m1 ). Note that T = T1 + T2 . By the first part of the proof of lemma 3.6 ε we immediately see that T1 < . 16 P 1 < We also know (by the definition of m1 ), that i≥n+m1 α1 (β 1 )α2 (β 1 ) . . . αi−1 (β 1 ) log αi (β 1 ) ε . Denote the tolal increase of the second part by T2+ ≥ 0, and the total decrease by 16− T2 ≥ 0. Then T2 = T2+ + T2− . Note that the total increase of the formula is bounded ε 1 by 2ε, and the decrease in the first part is less than , hence T2+ − T2− < 2 ε. On 16 16 2 ε + − . So T2 < 2 ε, and the other hand, all the terms are nonnegative, hence T2 < 16 16 3 ε 3 + − < 3ε, which completes the proof. T2 = T2 + T2 < 2 ε. Finally, T = T1 + T2 < 2 ε + 16 16 16 The following Lemma follows by the same argument as Lemma 3.6 by taking N ≥ 1 to be an arbitrary real number, not necessairily an integer. Lemma 3.14. For a β 1 as above, for any ε > 0 there is an m0 > 0, such that for any m ≥ m0 , and for any tail I = [an+m , an+m+1 , . . .] if we denote β I = [a1 , a2 , . . . , an , 1, 1, . . . , 1, an+m , an+m+1 , . . .],
then
X
α1 (β 1 )α2 (β 1 ) . . . αi−1 (β 1 ) log
i≥n+m
and
1 < ε, αi (β 1 )
n+m−1 X i=1
α1 (β I ) . . . αi−1 (β I ) log 1 − α1 (β 1 ) . . . αi−1 (β 1 ) log 1 < ε. αi (β I ) αi (β 1 )
We will also need the following Lemma in the proof of the Main Theorem.
Lemma 3.15. Let α = [a0 , a1 , a2 , . . .] and let ε > 0 be given. Then there is an N = N(ε) such that for any n ≥ N we have Φ(αn ) < Φ(α)+ε, where αn = [a0 , a1 , . . . , an−1 , 1, 1, 1, . . .]. The proof is not hard and is similar to the proof of Lemma 3.6. The influence on the “head” is small because the switch to 1’s occurs far away, and the “tail” does not increase too much because it can be shown to be very small (< 4ε ) after the change with n large enough. We leave the details to the reader. 4. Proving the Main Theorem We are now ready to prove the Main Theorem.
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M. BRAVERMAN, M. YAMPOLSKY
Theorem 4.1. There exists a parameter value θ ∈ R/Z such that the Julia set of the quadratic polynomial fθ (z) = z 2 + e2πiθ z is not computable. We note the the Main Theorem follows from Theorem 4.1 in an elementary way. Indeed, let θ ∈ R/Z be as promiced by Theorem 4.1. Then given an oracle access to θ it is easy to compute λ = e2πiθ , and vice versa. This implies that we could simulate the oracle for λ given an oracle for θ, and thus it is impossible to compute J(z 2 + λz) with an oracle access to λ. Setting c = λ2 /4 − λ/2 and Tc (z) = z − 2λ, we have J(z 2 + λz) = Tc (J(z 2 + c)).
Hence being able to compute J(z 2 +c) with an oracle access to c would allow us to compute J(z 2 + λz) with an oracle access to λ. Proof of Theorem 4.1. There are countably many oracle Turing Machines, hence we can enumerate them as M1φ , M2φ , . . .. We prove the theorem by constructing a number c that “fools” all the oracle Turing Machines M1φ , M2φ , . . . which are trying to compute Jc = J(fc ). We proceed iteratively, so on step i we “fool” the first i machines Mkφ . We then take the process to the limit, obtaining a number on which none of the machines works. Recall that r(θ) is the conformal radius of the Siegel disk associated with the polynomial Pθ (z) = z 2 + e2πiθ z, or zero, if θ is not a Brjuno number. On each iteration i we maintain an initial segment Ii = [a0 , a1 , . . . , aNi ] and an interval [li , ri ] such that the following properties are maintained: (4.1)
ri = r(γi ), where γi = [Ii , 1, 1, . . .],
and (4.2)
For any β = [Ii , tNi +1 , tNi +2 , . . .] with r(β) ∈ [li , ri ], the machines M1φ , M2φ , . . . , Miφ fail on β.
Moreover, the intervals we construct are nested: [li , ri ] ⊂ [li−1 , ri−1 ], and the sequence Ii contains Ii−1 as the initial segment. We also keep track of the variation (4.3) X 1 1 α1 (γi+1 )α2 (γi+1 ) . . . αj−1 (γi+1 ) log Ti = − α (γ )α (γ ) . . . α (γ ) log 1 i 2 i j−1 i α (γ ) α (γ ) j i+1 j i i≥0
at each step of the process, in order to be able to pass to the limiting sequence [a1 , a2 , a3 , . . .] in the end. For the basis of induction, set I0 = [1], r0 = r(γ0 ) and l0 = r0 /2, where γ0 = [1, 1, 1, . . .]. Then for i = 0 condition (4.1) holds by definition and condition (4.2) holds because it is empty.
NON-COMPUTABLE JULIA SETS
23
The induction step. We now have the conditions (4.1) and (4.2) for some i and would like to extend them to i + 1. φ Consider the machine M = Mi+1 . Denote by S the set of points β = [Ii , tNi +1 , tNi +2 , . . .] with r(β) ∈ [li , ri ] on which M works correctly. Let R be the set of possible conformal radii on S. There are two possibilities: Case 1: There exist ε0 > 0 and m0 ∈ N such that for every β ∈ S of the form β = [Ii , 1, 1, . . . , 1, . . .] we have |ri − r(β)| > ε0 . | {z } m0
In this case, select 0 < ε ≤ ε0 such that ri − ε > li . Set
Ii+1 = [Ii , 1, 1, . . . , 1], li+1 = ri − ε, and ri+1 = ri . | {z } m0
Since γi+1 = [Ii , 1, 1, . . .] = γi , we have r(γi+1 ) = r(γi) = ri = ri+1 and the condition (4.1) is satisfied. Suppose β = [Ii+1 , tNi+1 +1 , tNi+1 +2 , . . .] with r(β) ∈ [li+1 , ri+1 ]. Ii+1 is an extension of Ii , φ and the machines M1φ , M2φ , . . . , Miφ fail on β by the induction assumption. Mi+1 fails on β because r(β) ∈ [li+1 , ri+1 ], and so r(β) ∈ / R. This shows (4.2) and completes the proof for this case. Note that in this case we have γi+1 = γi, and so Ti = 0. The complementary case is the main part of the argument: Case 2. For every ε > 0 and m ∈ N we can find β ∈ S starting with Ii followed by m ones so that ri − ε < r(β) ≤ ri
(4.4)
Choose the number m of 1’s to be such that Lemma 3.14 holds with ε = 2−i . Choose an ε such that ri − 3ε > li , and let β ∈ S as above satisfying ri − ε < r(β) ≤ ri . Denote log(ri − ε) − log(ri − 2ε) log(ri − 2ε) − log(ri − 3ε) −i ε0 = min > 0. , ,2 8 8
Theorem 2.11 says that θ 7→ Φ(θ) + log r(θ) extends to a continuous 1-periodic function on R. Denote this function by f (θ). f is continuous and periodic, hence it must be uniformly continuous, and there is a δ > 0 such that if |x − y| < δ then |f (x) − f (y)| < ε0 . By Theorem 2.4 the Julia sets of Pθ depend continuously on θ with respect to the Hausdorff metric for θ ∈ S. By Proposition 2.13 the conformal radius r(•) is continuous on S. Hence there is a δ0 > 0 such that |r(x) − r(β)| < ε whenever |x − β| < δ0 and x ∈ S.
We choose m large enough, so that for any ζ = [Ii , 1, 1, . . . , 1, . . .], |γi − ζ| < δ. | {z } m
Write β = [Ii , 1, 1, . . . , 1, tNi +1 , tNi +2 , . . .]. By Lemma 3.15, there is an N such that for | {z } m
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M. BRAVERMAN, M. YAMPOLSKY
any n ≥ N, βn = [Ii , 1, 1, . . . , 1, tNi +1 , . . . , tNi +n , 1, 1, . . .] satisfies Φ(βn ) < Φ(β) + ε0 . | {z } m
We can choose n ≥ N large enough so that for any x with the initial segment Ii′ = [Ii , 1, 1, . . . , 1, tNi +1 , tNi +2 , . . . , tNi +n ], |x − β| < δ/2 and |x − β| < δ0 . | {z } m
Start with ω0 = βn = [Ii′ , 1, 1, . . .]. We have |ω0 − β| < δ, hence |f (ω0) − f (β)| < ε0 . So
log r(ω0) = f (ω0) − Φ(ω0 ) > f (β) − ε0 − Φ(β) − ε0 = log r(β) − 2ε0 . By Lemma 3.1 we can extend ω0 to ω1 = [Ii1 , 1, 1, . . .] so that Φ(ω0 ) + 2ε0 < Φ(ω1 ) < Φ(ω0 ) + 4ε0 . Hence and
log(r(ω1 )) = f (ω1 ) − Φ(ω1 ) > f (ω0 ) − ε0 − Φ(ω0 ) − 4ε0 = log(r(ω0)) − 5ε0 , log(r(ω1 )) = f (ω1 ) − Φ(ω1 ) < f (ω0 ) + ε0 − Φ(ω0 ) − 2ε0 = log(r(ω0 )) − ε0 .
Hence log(r(ω0 )) − 5ε0 < log(r(ω1 )) < log(r(ω0 )) − ε0 . In the same fasion we can extend Ii1 to Ii2 , and obtain ω2 = [Ii2 , 1, 1, . . .] so that log(r(ω1)) − 5ε0 < log(r(ω2 )) < log(r(ω1 )) − ε0 . Recall that log(r(ω0 )) > log(r(β)) − 2ε0 > log(ri − ε) − 2ε0 ≥ log(ri − 2ε) + 6ε0 . Hence after finitely many steps we will obtain Iik and ωk = [Iik , 1, 1, . . .] such that log(ri − 3ε) + ε0 < log(r(ωk )) < log(ri − 3ε) + 6ε0 < log(ri − 2ε).
Choose Ii+1 = Iik , γi+1 = ωk , li+1 = li , and ri+1 = r(ωk ). We have li+1 < ri+1 < ri − 2ε. Condition (4.1) is satisfied by definition. Condition (4.2) is satisfied for the first i machines because [li+1 , ri+1 ] is a subinterval of [li , ri ], and Ii+1 is an extension of Ii . Condition (4.2) is satisfied fot the i + 1-st machine M, because M could work correctly on x = [Ii′ , . . .] only with r(x) ∈ [ri − 2ε, ri + ε], and [li+1 , ri+1 ] is disjoint from [ri − 2ε, ri + ε]. We now estimate the total variation Ti in the process described above. By Lemma 3.14 we know that X 1 α1 (γi)α2 (γi ) . . . αj−1 (γi ) log < 2−i , αj (γi ) j≥N +1 i
and
Ni X 1 1 α1 (γi+1 ) . . . αj−1 (γi+1 ) log < 2−i . − α1 (γi ) . . . αj−1 (γi ) log αj (γi+1 ) αj (γi ) j=1
Denote d = Φ(γi+1 ) − Φ(γi ). Refer to the first Ni terms as the “head terms”, and the rest of the term as the “tail terms”. Denote the total increase in head terms from γi to γi+1 by h+ , and the total decrease by h− . Similarly define t+ and t− . Then d = h+ − h− + t+ − t− , and Ti = h+ + h− + t+ + t− . By the second inequality we have h+ + h− < 2−i . By the first one we have t− < 2−i . Hence t+ ≤ d + h+ + h− + t− < d + 2 · 2−i ,
NON-COMPUTABLE JULIA SETS
25
and Ti < 2−i + 2−i + d + 2 · 2−i = d + 4 · 2−i . We now bound d. d = Φ(γi+1 ) − Φ(γi ) = f (γi+1 ) − log(r(γi+1)) − f (γi) + log(r(γi )) < ε0 + log(r(γi )) − log(r(γi+1 )) < 2−i + log(ri ) − log(ri+1 ).
Hence Ti < log(ri ) − log(ri+1 ) + 5 · 2−i . Overall we get a bound on the sum of all the Ti ’s: ∞ X i=0
Ti < log(r0 ) − log(l0 ) + 5 ·
∞ X i=1
2−i = log(r0 ) − log(l0 ) + 10 < ∞.
Passing to the Limit. The completion of the proof relies on the following Lemma: Lemma 4.2. Let Ii be the initial segment of γi as above. Then the sequence {Φ(γi )} converges to a limit L = lim Φ(γi ). i→∞
Denote by I the inductive limit of the segments above, and let γ be the number represented by I, then Φ(γ) = L. 1 , and bi = α1 (γ)α2 (γ) . . . αi−1 (γ) log αi1(γ) . Proof. Denote bij = α1 (γj )α2 (γj ) . . . αi−1 (γj ) log αi (γ j) Our first claim is that for any i, limj→∞ bij = bi . Note that on step j the position being modified is at least j, hence by Lemmas 3.3 and 3.5 we see that for any j > i, 1 log α (γ ) αi (γj ) log i j < 2i−j , and log < 2i−j+1 . 1 αi (γ) log αi (γ)
Hence for all j > i, i−1 X k−j log bij < 2 + 2i−j+1 < 2i−j + 2i−j+1 < 2i−j+2. bi k=1 bi,i+2 bij Hence limj→∞ bi = 1. We know that bi is finite since log bi < 1, and bi,i+2 is finite by the definition of Φ(γi+2 ). Finally, limj→∞ bij = bi . P P ∞ ∞ Denote aij = bi,j+1 − bij and ai0 = bi1 . Then i=1 |aij | is the sum of all the j=1 variations T discussed above, and has been shown to be bounded by a fixed constant. Hence P∞ P∞ P∞ P∞ i i=1 |aij | < ∞. Hence the sum converges absolutely. j=1 i=1 |aij | = Φ(γ1 ) + j=0 Pj−1 P We have k=0 aik = bij . Taking j → ∞ yields ∞ k=0 aik = limj→∞ bij = bi . Hence Φ(γ) =
∞ X i=1
bi =
∞ X ∞ X i=1 k=0
aik .
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M. BRAVERMAN, M. YAMPOLSKY
For any j, P both Φ(γj ) and Φ(γ ) are finite, hence Φ(γj+1 ) − Φ(γj ) = P Pj+1 ∞ ∞ ∞ b = (b − b ) = ij i=1 ij i=1 i,j+1 i=1 aij . Hence L = lim Φ(γj ) = Φ(γ1 ) + j→∞
∞ X j=1
(Φ(γj+1) − Φ(γj )) =
∞ X
bi1 +
i=1
∞ X ∞ X
aij =
j=1 i=1
P∞
i=1 bi,j+1
∞ X ∞ X
−
aij .
j=0 i=1
P P∞ By the absolute convergence of ∞ i=1 |aij | < ∞, the limit L above exists, and we can j=0 apply Fubini’s theorem to obtain ∞ ∞ X ∞ ∞ X X X aij = aij = L, Φ(γ) = i=1 j=0
j=0 i=1
which completes the proof.
Finalizing the argument. Let γ be the limit from the previous section. We claim the no machine works properly on γ. For any i, Ii is an initial segment of γ by definition. We only need to see that r(γ) ∈ [li , ri ] to show that Miφ fails on γ. We claim that (4.5)
r(γ) = lim ri . i→∞
r(γ) ∈ [li , ri ] follows, since li is nondecreasing and ri is nonincreasing. To show (4.5) observe that γ = limi→∞ γi , and by continuity of f , f (γ) = limi→∞ f (γi). Together with Lemma 4.2 we obtain lim log(ri ) = lim (f (γi ) − Φ(γi )) = lim f (γi ) − lim Φ(γi ) = f (γ) − Φ(γ) = log(r(γ)),
i→∞
i→∞
i→∞
i→∞
thus (4.5) follows, which completes the proof. Non-computability of Yoccoz’s Brjuno function. The proof of Theorem 2.14 requires very minor modifications to the preceding argument. We proceed by induction maintaining on iteration i an initial segment Ii and an interval [li , ri ] such that the following analogues of (4.1) and (4.2) are maintained: (4.6)
li = Φ(γi ), where γi = [Ii , 1, 1, . . .],
and (4.7)
For any β = [Ii , tNi +1 , tNi +2 , . . .] with Φ(β) ∈ [li , ri ],
the machines M1φ , M2φ , . . . , Miφ fail on β. The induction step is carried out similarly to the proof of the Main Theorem. The only difference is that now the roles of the left endpoint li and the right endpoint ri are interchanged. The reason for this switch is that by playing with the tail of γi we can decrease r(γi) by a small amount, and similarly we can increase Φ(γi ) by a small controlled amount. Case 1 of the proof is as trivial here as in the proof of the Main Theorem. Case 2 is actually simpler here, and can be done with elementary means, because we no longer need to apply Theorem 2.11, and the small growth of Φ is controlled here by the lemmas from §3.
NON-COMPUTABLE JULIA SETS
27
By keeping the same bounds on the variation one finalizes the argument exactly as in the proof of the Main Theorem. 5. Concluding remarks Let us outline here how the methods of [BC1] can be applied to prove Theorem 4.1 instead of the estimates of §3 (we note that a newer version of the same result exists [ABC], where the arguments we quote are simplified). The main technical result of that paper is the following. Let α = [a0 , a1 , . . .] be a Brjuno number, and as before denote pk /qk the sequence of its continued fraction approximants. Let A > 1 and for each integer n ≥ 0 set α[n] = [a0 , a1 , . . . , an , Aqn , 1, 1, 1, . . .]. Then for this particular sequence of Brjuno approximants of α, Φ(α[n]) −→ Φ(α) + log A, n→∞
and moreover, lim r(α[n]) = r(α)/A. The last equality can be used to construct the “drops” in the value of the conformal radius of the Siegel disk needed to inductively fool the TM’s. In this way, one obtains a sequence of Brjuno numbers θi → θ with conformal radii ri = r(θi ) > ri+1 such that lim ri = r > 0, and θi is not computable by all TM’s up to i-th. It remains to show that r(θ) = r, as a priori only the inequality “≤” is known. Buff and Ch´eritat demonstrate it in their context. The idea is roughly speaking in showing that the boundary of ∆(θi ) is well approximated by a periodic cycle of a high period. The perturbation θi 7→ θi+1 is then chosen sufficiently small so that the cycle does not move much. As a final remark, let us point out: Remark 5.1. Combining the methods of [BC1] with our argument as outlined here one may strengthen Theorem 4.1 by showing that there exists a non-computable Siegel Julia set for which the boundary of the Siegel disk is smooth.
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References [BCSS] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation, Springer, New York, 1998. [ABC] A. Avila, X. Buff, A. Ch´eritat. Siegel disks with smooth boundaries. Preprint. [BC1] X. Buff, A. Ch´eritat, Quadratic Siegel disks with smooth boudaries. Preprint Univ. Paul Sabatier, Toulouse, III, Num. 242. [BC2] X. Buff, A. Ch´eritat, The Yoccoz Function Continuously Estimates the Size of Siegel Disks, Preprint, 2003. [Bru] A.D. Brjuno. Analytic forms of differential equations, Trans. Mosc. Math. Soc 25(1971) [Brv] M. Braverman, “Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are PolyTime Computable”, Thesis, University of Toronto, 2004, and Proc. CCA 2004, to appear. [Do] A. Douady. Does a Julia set depend continuously on the polynomial? In Complex dynamical systems: The mathematics behind the Mandelbrot set and Julia sets. ed. R.L. Devaney, Proc. of Symposia in Applied Math., Vol 49, Amer. Math. Soc., 1994, pp. 91-138. [DH1] A. Douady, J.H. Hubbard. Etude dynamique des polynˆomes complexes, I-II. Pub. Math. d’Orsay, 1984. ´ Norm. [DH2] A. Douady, J.H. Hubbard. On the dynamics of polynomial-like mappings. Ann. Sci. Ec. Sup., 18(1985), 287-343. [Ko1] K. Ko, Complexity Theory of Real Functions, Birkh¨auser, Boston, 1991. [Ko2] K. Ko, Polynomial-time computability in analysis, in ”Handbook of Recursive Mathematics”, Volume 2 (1998), Recursive Algebra, Analysis and Combinatorics, Yu. L. Ershov et al. (Editors), pp 1271-1317. [MMY] S. Marmi, P. Moussa, J.-C. Yoccoz, The Brjuno functions and their regularity properties, Commun. Math. Phys. 186(1997), 265-293. [Mat] Y. Matiyasevich, Hilbert’s Tenth Problem, The MIT Press, Cambridge, London, 1993. [McM1] C. McMullen. Complex dynamics and renormalization. Annals of Math. Studies, v.135, Princeton Univ. Press, 1994. [Mil] J. Milnor. Dynamics in one complex variable. Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999. [Pom] C. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, 1992. [RW] R. Rettinger, K. Weihrauch, The Computational Complexity of Some Julia Sets, in STOC’03, June 9-11, 2003, San Diego, California, USA. [Ret] R. Retinger. A fast algorithm for Julia sets of hyperbolic rational functions, Proc. of CCA 2004, to appear. [Sie] C. Siegel, Iteration of analytic functions. Ann. of Math. (2) 43, (1942). 607–612 [Sip] M. Sipser, Introduction to the Theory of Computation, PWS Publishing Company, 1997. [Tur] A. M. Turing, On Computable Numbers, With an Application to the Entscheidungsproblem. In Proceedings, London Mathematical Society, 1936, pp. 230-265. [Wei] K. Weihrauch, Computable Analysis, Springer, Berlin, 2000. [Yoc] J.-C. Yoccoz, Petits diviseurs en dimension 1, S.M.F., Ast´erisque, 231(1995).
