Nonlocally maximal and premaximal hyperbolic sets - BYU Math ...
NONLOCALLY MAXIMAL AND PREMAXIMAL HYPERBOLIC SETS T. FISHER, T. PETTY, AND S. TIKHOMIROV Abstract. We prove that for any closed manifold of dimension 3 or greater there is an open set of smooth flows that have a hyperbolic set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a hyperbolic set is an intrinsic property for premaximality. Lastly, we review some results due to Anosov that concern premaximality.
1. Introduction Since the 1960s the study of hyperbolic sets has been a cornerstone in the field of dynamical systems. These sets are remarkable not only in their complexity, but also in the fact that they persist under perturbations. Additionally, for a point in a hyperbolic set the derivative of the map at this point gives information on the local dynamics for the original nonlinear map. As a reminder, for a diffeomorphism f : M → M , a compact invariant set Λ is hyperbolic for f if TΛ M = Es ⊕ Eu is a Df -invariant splitting such that Es is uniformly contracted and Eu is uniformly expanded by Df . Anosov was one of the pioneers in studying hyperbolic sets. Indeed, if the entire manifold is a hyperbolic set for a diffeomorphism, then the diffeomorphism is called Anosov. This is one of the best understood classes of hyperbolic sets. 2000 Mathematics Subject Classification. 37D20, 37D05, 37C05. Key words and phrases. hyperbolic sets, hyperbolic flow, premaximal, locally maximal, isolated. The authors would like to thank the Sixth International Conference on Differential and Functional Differential Equations during which much of the paper was prepared. T.F. is supported by Simons Foundation grant # 239708. S. T. is partially supported by Chebyshev Laboratory under Russian Federation Government grant 11.G34.31.0026, JSC “Gazprom neft”, Saint Petersburg State University research grant 6.38.223.2014, Russian Foundation of Basic Research 15-01-03797a and German-Russian Interdisciplinary Science Center (G-RISC). 1
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Another important class of hyperbolic sets are those that are locally maximal. For a compact metric space X and a continuous homeomorphism T : X → X a set K ⊂ X is locally maximal if there exists a neighborhood U of K such that \ K= T n (U ). n∈Z
Hence, such sets are the maximal invariant set within U . Locally maximal sets were defined more or less simultaneously by Anosov and Conley (who called these sets isolated) in the 1960s. Alexseev proved in [1] that a shift space is locally maximal if and only if it is a shift of finite type. Furthermore, he proved that given any shift space Σ and a neighborhood U of Σ there exists a locally maximal shift Σ0 containing Σ and contained in U . A question that was posed in the 1960s by Anosov, Alexseev, and others (that is stated for instance in [15, p. 272]), is the following: Question 1.1. If Λ is a hyperbolic set and U is a neighborhood of Λ, ˜ such that Λ ⊂ Λ ˜ ⊂ U? then is there a locally maximal hyperbolic set Λ As stated by Anosov in [3] it was hoped at the time that the answer would be in the affirmative. One reason this was hoped for is that locally maximal hyperbolic sets are more easily classified. Indeed, a hyperbolic set is known to be locally maximal if and only if it has a local product structure (defined in Section 2). A standard assumption used in characterizing topological and/or measure theoretic properties of hyperbolic sets is that the set is locally maximal. For instance, Smale’s Spectral Decomposition Theorem, see for instance [15, p. 575], is valid for locally maximal hyperbolic sets. In fact, quite frequently even when this is not specifically stated in a theorem one finds in the proof that this assumption is required for the result to hold. 1.1. Nonlocally maximal hyperbolic sets. It was shown by Crovisier in [9] that there is a hyperbolic set on the 4-torus that is never included in a locally maximal set. Later, in [13] it was shown that any compact boundaryless manifold with dimension greater than or equal to 2 has a C r open set of diffeomorphisms where 1 ≤ r ≤ ∞ such that each diffeomorphism in the open set contains a hyperbolic set that is not included in a locally maximal one. Our first result is an extension of the results in [13] to hyperbolic flows. As a reminder, for a smooth flow φ : R × M → M , a compact φ-invariant set Λ is hyperbolic for φ if TΛ M = Es ⊕ Ec ⊕ Eu is a flow invariant splitting such that Es is uniformly contracted, Ec is the flow
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direction, and Eu is uniformly expanded. A set Λ is locally maximal for the flow φ if there is an open set U containing Λ such that Λ = ∩t∈R φt (U ). Theorem 1.2. Let M be a compact, boundaryless C r manifold for 1 ≤ r ≤ ∞ with dim M ≥ 3 and X k (M ) be the set of C k flows on M where 1 ≤ k ≤ r. Then there exists a C k open set of flows on M such that each flow contains a hyperbolic set not contained in a locally maximal one. Remark 1.3. We notice the following. 1. If dim M ≤ 2 then every hyperbolic set for a smooth flow is a finite union of hyperbolic closed trajectories and hence it is locally maximal. 2. Also, one can always suspend the map constructed in [13] and have a topological flow. However, it remains to be seen if the suspension will still be smooth. Also, the suspension could introduce nontrivial topology and it may not be possible to obtain the result of 1.2 on any manifold of dimension 3 or larger. As in [13, Theorem 1.5] we can show the following result. Theorem 1.4. Let Λ be a hyperbolic set for a flow and U be a neighborhood of Λ, then there exists a hyperbolic set Λ0 with a Markov partition for the flow such that Λ ⊂ Λ0 ⊂ U . The proof is very similar to that in [13]. Indeed, the necessary theorems used in [13] to prove the similar results for maps hold for hyperbolic sets for flows, and hence the proof is left to the reader. The hyperbolic set constructed in Theorem 1.2 above need not transitive under the flow. However, as in [13, Theorem 1.6] one can construct a flow in higher dimensions with a transitive hyperbolic set that is not contained in a locally maximal one. 1.2. Premaximality. In this paper we also examine conditions under which a hyperbolic set, Λ, is included in a locally maximal hyperbolic set within an arbitrarily small neighborhood of Λ. Following the terminology introduced by Anosov in [3] we define a hyperbolic set Λ for a diffeomorphism to be premaximal if for any open ˜ such set U containing Λ there is a locally maximal hyperbolic set Λ ˜ ⊂ U . In [3] Anosov proves that any zero-dimensional that Λ ⊂ Λ hyperbolic set for a diffeomorphism is premaximal, and in [2] Anosov proves there is an intrinsic property for premaximal hyperbolic sets for diffeomorphisms. Moreover the following holds.
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Theorem 1.5. [6] Let f : M → M and f 0 : M 0 → M 0 be diffeomorphisms, Λ a hyperbolic set for f , Λ0 a hyperbolic set for f 0 , and h : Λ → Λ0 a homeomorphism such that h ◦ f = f 0 ◦ h. If U is a neighborhood of Λ and U 0 is a neighborhood of Λ0 , then there exists neighborhoods V ⊂ U of Λ and V 0 ⊂ U 0 of Λ0 and continuous injective equivariant maps h1 : If (U ) → M 0 and h2 : If 0 (U 0 ) → M such that h1 |Λ = h, and h1 (If (V )) ⊂ If 0 (U 0 ),
h2 (If 0 (V 0 )) ⊂ If (U ), f ◦ h2 |If 0 (V 0 ) = h2 ◦ f 0 |If 0 (V 0 ) ,
h1 ◦ f |If (V ) = g ◦ h1 |If (V ) , h2 ◦ h1 |If (V ) = id, and
h1 ◦ h2 |If 0 (V 0 ) = id.
The above theorem shows that f |Λ defines the set of trajectories that lie in a sufficiently small neighborhood of Λ. However, in [6] the specific intrinsic property for premaximality is not stated. We extend result of [6] to the case of flows and prove the premaximality is an intrinsic property for hyperbolic sets for flows. Let X and X 0 be vector fields on smooth compact Riemannian manifolds M and M 0 respectively. Denote by φ and φ0 flows generated by them. An increasing homeomorphism of the real line α : R → R is called a reparametrization. Let Λ and Λ0 be hyperbolic sets for X and X 0 respectively. We say that Λ and Λ0 are topologically equivalent if there exists a homeomorphism h : Λ → Λ0 and a continuous map α : M × R → R such that h ◦ φt (x) = φ0 (α(x, t), h(x)),
x ∈ Λ, t ∈ R
where α(x, ·) is a reparametrization for each x ∈ Λ. In this case there exists a continuous map β : M 0 × R → R, such that β(h(x), α(x, t)) = t,
x ∈ Λ, t ∈ R
α(h−1 (x0 ), β(x0 , t0 )) = t0 , and x0 ∈ Λ0 , t0 ∈ R. Theorem 1.6. Let Λ and Λ0 be hyperbolic sets for vector fields X and X 0 respectively. Assume that Λ and Λ0 are topologically equivalent. Then Λ is premaximal if and only if Λ0 is premaximal. Below we provide an equivalent condition to premaximality for diffeomorphisms and flows. Before stating the result we define some important terms involving shadowing. Let Φ = φ(t, x) be a dynamical system where t can be taken to be discrete or continuous. If t is discrete, we assume the dynamical system is generated by a diffeomorphism of a compact manifold to itself. If t is continuous, we assume that the dynamical system is generated by a smooth vector field on a compact
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manifold. Let a > 0 be an expansivity constant for some neighborhood of a hyperbolic set Λ and let δ0 > 0 be such that any δ0 -pseudo orbit in Λ can be a/2-shadowed by an exact trajectory (definitions are given in the next section). Note that due to expansivity the shadowing trajectory is unique (for the case of a flow this is true up to a reparametrization). For Λ a hyperbolic set for a diffeomorphism and δ ∈ (0, δ0 ) the shadowing closure (or δ-shadowing closure) of Λ is sh(Λ, δ) = {y ∈ M : y shadows a δ-pseudo orbit in Λ}. For a fixed δ > 0 we can construct a sequence of shadowing closures Λ0 , Λ1 , ..., where Λ0 = Λ and Λj = sh(Λj−1 , δ) for j ∈ N. We say a shadowing sequence stabilizes if Λj = Λj+1 for all j ≥ N where N ∈ N. Theorem 1.7. For a hyperbolic set Λ of a dynamical system Φ the following statements are equivalent (1) Λ is premaximal; (2) for any neighborhood U of Λ the shadowing closure stabilizes inside U for some δ > 0. Note that due to Theorem 1.5 and Theorem 1.6 the second property in Theorem 1.7 for diffeomorphisms is intrinsic. The paper proceeds as follows. In Section 2 we review relevant background on hyperbolicity and flows. In Section 3 we prove Theorem 1.2. In Section 4 we review the results of Anosov in [2, 3, 4, 5, 6] and prove Theorems 1.6 and 1.7. 2. Background 2.1. Hyperbolic sets for flows. We first review properties of hyperbolic sets for flows. Definition 2.1. Let X be a metric space and φ a continuous flow on X. Then for x ∈ X we define the stable set W s (x) := {y ∈ X : lim d(φt (x), φt (y)) = 0}. t→∞
Further, for ε > 0 the ε-stable set is Wεs (x) := {y ∈ W s (x) : d(φt (x), φt (y)) ≤ ε for all t ≥ 0}. Note that the unstable sets W u (x) and Wεu (x) are defined identically under the flow φ−t . Furthermore, we define the center-stable set [ W cs (x) := {φt (W s (x))|t∈R } = W s (y). y∈φt (x) t∈R
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The center-unstable set of x is defined to be the center-stable set of x s under φ−t . We will also use the notation Wloc to mean Wεs for sufficiently u similarly to mean Wεu for small ε (depending on the context) and Wloc small ε. Let X be a manifold and φ be a C r flow on X. If Λ is a hyperbolic set for φ and p ∈ Λ, then W s (p) is a C r immersed submanifold and is an immersed copy of Rk where k = dim Es (x). Similar statements hold for the unstable sets, center-stable sets, and center-unstable sets. Also note that the stable and unstable manifolds vary continuously on the point p. Definition 2.2. For a metric space X and a flow φ, a set Γ ⊂ X is said to have a local product structure if for all ε > 0 sufficiently small there exists a δ > 0 such that given x, y ∈ Γ with d(x, y) < δ we have, for some real |t| < ε, a unique point S(x, y) := b ∈ Wεu (φt (x)) ∩ Wεs (y) contained in Γ. Remark 2.3. Note that for any hyperbolic set Λ there always exist constants δ and ε sufficiently small such that x, y ∈ Λ and d(x, y) < δ implies S(x, y) = Wεu (φt (x)) ∩ Wεs (y) is a unique point in the manifold, but may not be in Λ. The following lemma is also critical to the paper. Note that this lemma is almost always stated and proved for maps, but is in fact true for flows as well (see [8] and [16]). Lemma 2.4. A hyperbolic set Γ has a local product structure if and only if it is locally maximal. We also need the notion of the shadowing property. For δ > 0 a map g : R → M is an δ-pseudo orbit if the following holds d(g(t + τ ), φτ (g(t))) < δ,
t ∈ R, |τ | < 1.
A δ-pseudo orbit g is ε-shadowed by a point x0 if there exists a reparametrization α : R → R satisfying d(g(t), φα(t) (x0 )) < ε, and α(t1 ) − α(t2 ) < ε for t1 6= t2 . − 1 t1 − t2 A vector field X is expansive on a compact metric space W if there exist constants a, τ0 > 0 such that if x1 , x2 ∈ W and there exists a reparametrization α : R → R such that the following inequalities hold d(φ(α(t), x1 ), φ(t, x2 )) < a, then x2 = φ(τ, x1 ), where τ ∈ (−τ0 , τ0 ).
t ∈ R,
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Theorem 2.5. Let Λ be a hyperbolic set for a flow φ on a compact manifold M . Then the following hold: • there exists a neighborhood W of Λ such that φ is expansive on \ Iφ (W ) := φt (W ); t∈R
and • there exists a neighborhood U (Λ) ⊂ W such that for any ε > 0 there exists δ > 0 such that any δ-pseudo orbit g ⊂ U can be ε-shadowed by some point x0 ∈ M . Definition 2.6. For φ a flow on a compact metric space X a nonempty subset A of X is called an attractor if it satisfies the following three conditions. (i) A is forward-invariant under φ; i.e., x ∈ A implies φt (x) ∈ A for all t > 0. (ii) There exists a neighborhood of A, called the basin of attraction of A and denoted B(A), which consists of all points that tend towards A under φt as t → ∞. In other words, B(A) consists of all points x such that for any open neighborhood N of A, there exists T > 0 such that φt (x) ∈ N for all t > T . (iii) No proper subset of A satisfies conditions (i) and (ii). When an attractor Λ is (uniformly) hyperbolic we know that • periodic points are dense in Λ, • x ∈ Λ implies W cu (x) ⊂ Λ, and S • for any periodic point x ∈ Λ we know y∈O(x) W cs (y) is dense in B(Λ). We will need the following technical result, known in the literature as the Inclination Lemma, or λ-lemma. The statement can be found in [7]. Note that the statement for hyperbolic periodic points would be similar. Lemma 2.7 (Inclination Lemma). Let p ∈ M be a hyperbolic fixed point for a C r flow φ, for r ≥ 1, with local stable and unstable manifolds s u u Wloc (p) and Wloc (p), respectively. Fix an embedded disk B in Wloc (p) u which is a neighborhood of p in Wloc (p), and fix a neighborhood V of s this disk in M. Let D be a transverse disk to Wloc (p) at a point z such that D and B have the same dimension. Write Dt for the connected component of φt (D) ∩ V which contains φt (z), for t ≥ 0. Then, given ε > 0 there exists T > 0 such that for all t > T the disk Dt is ε-close to B in the C r -topology.
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2.2. Hyperbolic sets of diffeomorphisms. Since the statement of Theorem 1.7 refers to diffeomorphisms as well as smooth flows we now review some definitions for discrete dynamical systems. For f : X → X a homeomorphism of a metric space and δ > 0 an δ-pseudo orbit is a sequence {xj }m l where • l ∈ {−∞} ∪ Z, m ∈ Z ∪ {∞}, l < m, and • d(f (xj ), xj+1 ) < δ for all j ∈ [l, m]. For a δ-pseudo orbit {xj }m l we say this sequence is ε-shadowed by a j point x ∈ X if d(f (x), xj ) < ε for j ∈ [l, m]. We say that a homeomorphism f of a compact metric space W is expansive if there exists a constant a > 0 such that if \ f n (W ) x1 , x2 ∈ If (W ) := n∈Z
and d(f n (x1 ), f n (x2 )) < a,
n ∈ R,
then x2 = x1 . Theorem 2.8. (Shadowing Lemma) Let Λ be a hyperbolic set for f : M → M a diffeomorphism. Then there exists neighborhood U (Λ) such that for all ε > 0 there exists an δ > 0 such that if {xj }∞ −∞ ⊂ U is an δ-pseudo orbit, then there exists x ∈ M that ε-shadows {xj }∞ −∞ . Let f : M → M be a diffeomorphism and Λ be a hyperbolic set for f . For ε > 0 sufficiently small and x ∈ Λ the local stable and unstable manifolds and stable and unstable manifolds are defined similar to the case for flows and for a C r diffeomorphism f the stable and unstable manifolds of a hyperbolic set are C r injectively immersed submanifolds. For Λ a hyperbolic set we know that if ε is sufficiently small and x, y ∈ Λ, then Wεs (x) ∩ Wεu (y) consists of at most one point. For such an ε > 0 define Dε = {(x, y) ∈ Λ × Λ | Wεs (x) ∩ Wεu (y) ∈ Λ} and [·, ·] : Dε → Λ so that [x, y] = Wεs (x) ∩ Wεu (y). We will also need openness of hyperbolicity. Lemma 2.9. Let Λ ⊂ M be a hyperbolic set of the diffeomorphism f : U → M . Then for any open neighborhood V ⊂ U of Λ and every δ > 0 there exists ε > 0 such that if f 0 : U → M and dC 1 (f |V , f 0 ) < ε there is a hyperbolic set Λ0 = f 0 (Λ0 ) ⊂ V for f 0 and a homeomorphism h : Λ0 → Λ with dC 0 (Id, h)+dC 0 (Id, h−1 ) < δ such that h◦f 0 |Λ0 = f |Λ ◦h. Moreover, h is unique when δ is sufficiently small.
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2.3. Normal hyperbolicity. For embedding constructions into higher dimensions, we will need the notion of normal hyperbolicity. A normally hyperbolic invariant manifold (NHIM) is a generalization of a hyperbolic fixed point and a hyperbolic set. Fenichel proved that NHIMs and their stable and unstable manifolds are persistent under perturbation [11], [12]. We define NHIMs for maps, but the definition for flows is similar (and more technical). Definition 2.10. Let M be a compact smooth manifold and f : M → M a diffeomorphism. Then an f -invariant submanifold Λ of M is said to be a normally hyperbolic invariant manifold if there exist m ∈ N such that the mapping f m satisfies the following property. There exists a continuous invariant bundle TΛ M = E s ⊕ T Λ ⊕ E u (x), and continuous positive functions ν, νˆ, γ, γˆ : M → R such that ν, νˆ < 1,
ν < γ < γˆ < νˆ−1
and for all x ∈ Λ, v ∈ Tx M , |v| = 1 |Df m (x)v| ≤ ν(x), v ∈ E s (x); γ(x) ≤ |Df m (x)v| ≤ γˆ (x), v ∈ T Λ; |Df m (x)v| ≥ νˆ−1 (x), v ∈ E u (x). Adapting the above for flows gives us an important result ([10, p. 215]) which says that if a C r vector field Y in some C 1 neighborhood of our original vector field X (equated with a flow φ, under which Λ is invariant) there is a C r manifold ΛY invariant under Y and C r diffeomorphic to Λ. An immediate consequence of this is that the dynamics on ΛY under the vector field Y are a perturbation of the dynamics of Λ under X. 3. Nonlocally maximal sets for flows The foundation of the proof of Theorem 1.2 is the Plykin attractor, see for instance [15, p. 537-41] for a construction of the Plykin attractor. The first author used this map with some modifications to prove Theorem 1.3 in [13] on the existence of hyperbolic sets not included in locally maximal ones. To extend the results of [13] to flows we need to show the suspension can be done smoothly and in such a way that no nontrivial topology is introduced in the suspension. In [14] Hunt shows that the Plykin attractor can be suspended smoothly in such a way that the suspended flow is on the solid 2torus. The result of the construction is a smooth flow of a solid 2-torus where the basin of attraction for the suspended Plykin attractor is the
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interior of the torus. Using that flow, we apply the construction in [13] to Hunt’s flow and address the technicalities. Proof of Theorem 1.2. We now show how to adapt the construction in [13] to the suspension flow described above. First we would like to sketch the main steps of the construction of the example for a more detailed proof see [17]. Let T1 be the solid 2-torus used in the construction described above. We first embed Hunt’s Plykin attractor in a slightly larger solid 2-torus, T , so that the flow is the identity in a neighborhood of the boundary and then extend the flow to a 3-disk, D, so that it is the identity on D − T . This will allow us to embed the example into any arbitrary 3-manifold. We now modify the flow on T1 to incorporate the construction in [13]. Let Λa be the Plykin attractor and φ be the flow on D. Fix some p ∈ ∂T1 . Take a sufficiently small open neighborhood U of O(p), small enough to be disjoint from ∂T and Λa , and alter φ in U so that p is a hyperbolic saddle periodic point with W s (p) ⊂ U ∩ ∂T1 . Also, W cu (p) \ O(p) contains two components. One of which is contained in T − T1 and the other, denoted W ∗ (p) is contained strictly in the interior of T1 . Let q be a periodic point in Λa . Since W cs (q) = W s (Λa ) = int(T1 ) we know that given any point of W ∗ (p) that there must exist some point in W cs (q) arbitrarily close to it. Fix z ∈ W ∗ (p). Perturb the flow in a neighborhood of z so that z ∈ W cs (q) t W ∗ (p). This can be done since z is a wandering point for the flow. Here we will need two definitions. A hyperbolic set Λ for a C 1 flow has a heteroclinic tangency if there exist x, y ∈ Λ such that W s (x) ∩ W u (y) contains a point of tangency. A point of quadratic tangency for a C 2 flow is defined as a point of heteroclinic tangency where the curvature of the stable and unstable manifolds differs at the point of tangency. Now after a further perturbation to the flow as in [13] there exists a point w ∈ W u (z) and a point q 0 ∈ W u (q) such that W u (z) and W s (q 0 ) have a quadratic tangency at w. Let I be the segment of W u (p) from z to w, and let J be the segment of W u (q) from q to q 0 . The resulting flow will contain a hyperbolic set that cannot be contained in a locally maximal set as we show below. Figure 1 demonstrates a cross section of the constructed flow. Let Λ = Λa ∪ O(p) ∪ O(z). Standard arguments as in [13], adapted to flows, show that this is a hyperbolic set under φ. Now suppose
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w
z
q
J
q‘
Figure 1. Intervals I and J. Λ ⊂ Λ0 , where Λ0 is a locally maximal hyperbolic set. Modifying the proof in [13] we see from the product structure on Λ0 that this implies w ∈ Λ0 , a contradiction. Hence, Λ is not contained in a locally maximal hyperbolic set. We now show the construction is robust under perturbation. Since transversality is trivially open, and hyperbolicity is open by Lemma 2.9, it is sufficient to show that there remains a point w˜ ∈ W u (˜ p) ∩ W s (˜ u) cu for some u˜ ∈ Wloc (˜ q ). Let p˜ and q˜ be the continuations of p and q for the perturbed flow. By construction, the stable manifolds for all cu the x ∈ Wloc (˜ q ) locally foliate the region, so there must exist a point cu q ) and a point w˜ ∈ W cs (˜ u) ∩ W u (˜ p) such that the oneu˜ ∈ Wloc (˜ u dimensional path W (˜ p) remains tangent to the two-dimensional plane W cs (˜ u) at w. ˜ Specifically, we have Tw˜ W u (˜ p) ( Tw˜ W cs (˜ u). Using normal hyperbolicity we can embed our example into any smooth manifold M of dimension greater than 3. Furthermore, normal hyperbolicity implies the construction is still robust in this setting. 2 4. Premaximality Before we proceed to the proof of Theorems 1.6 and 1.7 let us review results by Anosov on premaximality [2, 3, 4, 5, 6]. As was mentioned in Section 1, Theorem 1.5 implies that premaximality is an intrinsic property.
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Let us recall the main idea of the proof of Theorem 1.5. For δ > 0 denote P (δ) as the set of δ-pseudo orbits {yn ∈ Λ}n∈Z endowed with the Tikhonov product topology. Consider the shift map σ : P (δ) → P (δ) defined as σ({yn }) = {yn+1 }. The Shadowing Lemma implies that for any ε > 0 there exists δ > 0 such that for any {yn } ∈ P (δ) there exists point x such that (1)
d(yn , f n (x)) < ε.
Note that due to the expansivity property we know for small enough ε > 0 that such a point x is unique. Fix such an ε > 0 and a corresponding δ from the shadowing lemma. Consider the map T : P (δ) → M defined by the condition that for {yn } ∈ P (δ) the point x = T ({yn }) is the unique point satisfying (1). It is easy to show that (2)
T ◦ σ = f ◦ T.
Now let us consider c > 0 and a small neighborhood U ⊂ B(c, Λ) of Λ and a point z such that O(z) ⊂ U . There exists a sequence of points {yn } satisfying (3)
d(yn , f n (z)) < c.
For any δ > 0 there exists a c > 0 such that inequality (3) implies {yn } ∈ P (δ) and (4)
T ({yn }) = z.
We can similarly define for Λ0 and f 0 sets P 0 (δ) and maps T 0 , σ 0 . Note that similarly to (2) the equality (5)
T 0 ◦ σ0 = f 0 ◦ T 0
holds. For any δ10 > 0 there exists δ1 > 0 such that (6)
h(P (δ1 )) ⊂ P 0 (δ10 ).
(Recall that h : Λ → Λ0 is a conjugacy between Λ and Λ0 .) Similarly for any δ2 there exists δ20 such that (7)
h−1 (P 0 (δ20 )) ⊂ P (δ2 ).
Equations (2), (4), (5), (6), and (7) allow us to conclude Theorem 1.5. For a detailed exposition of the proof we refer the reader to the original paper [6]. To study premaximality of the zero-dimensional hyperbolic sets we need the following notion. Let A be an “alphabet”: A = {1, . . . , n}
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and Ω = AZ equipped with the Tikhonov topology and metric X 1 d(a, b) = I(ai , bi ), |i| 2 i∈Z where a = (ai ), b = (bi ) ∈ Ω and I(ai , bi ) equal to 0 if ai = bi and 1 otherwise. Consider the shift map σ = Ω → Ω defined as (σ(a))i = ai+1 . Consider some set W of admissible words of lengths k ≥ 1 in the alphabet A and consider MW ⊂ Ω such that all subwords of all a ∈ MW of length k are admissible. Theorem 4.1. [1] Consider a set Λ ⊂ Ω. The following holds (1) the set Λ is locally maximal for σ if and only if there exists k ≥ 1 and set of admissible words W such that Λ = MW ; (2) the set Λ is premaximal. In [3] it was proved that zero-dimensional hyperbolic sets are topologically conjugated to Bernoulli shifts, which implies the next result. Theorem 4.2. Let Λ be a zero-dimensional hyperbolic set of a diffeomorphism f . Then Λ is premaximal. Burns and Gelfert were able to extend the above result to prove that a 1-dimensional hyperbolic set for a flow is premaximal [8, Proposition 8]. The proof follows an argument provided by Anosov after personal communications. In [4] Anosov obtain the following sufficient condition for a hyperbolic set to not be premaximal. Theorem 4.3. Let Λ be a hyperbolic set of f ∈ C 1 . Assume that there exists a family of exact trajectories ξ : Z × [0, a] → M such that (1) ξn+1,t = f (ξ(n, t)), (2) ξ(0, 0) ∈ Λ, (3) d(ξ(n, t), Λ) → 0 uniformly in t as |n| → ∞, and (4) there exists t1 ∈ [0, a] such that ξ(0, t1 ) ∈ / Λ. Then Λ is not premaximal. Note that the examples of Crovisier and Fisher satisfy these conditions. We now prove an analog of Theorem 1.5 for flows from which Theorem 1.6 will follow. Theorem 4.4. Let Λ and Λ0 be hyperbolic sets for smooth flows φ and φ0 respectively. Assume that Λ and Λ0 are topologically equivalent, with the corresponding map h : Λ → Λ0 . If U is a neighborhood of Λ and U 0 is a neighborhood of Λ0 , then there exists neighborhoods V ⊂ U of Λ
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and V 0 ⊂ U 0 of Λ0 , numbers τ0 , τ00 > 0, and (not necessarily continuous) maps h1 : Iφ (V ) → M 0 and h01 : Iφ0 (V 0 ) → M such that h1 |Λ = h and h01 |Λ0 = h−1 .
(8)
Furthermore, for any ε > 0 there exists δ > 0 such that if x1 , x2 ∈ Iφ (V ) and d(x1 , x2 ) < δ, then there exists |τ 0 | < τ00 such that d(h1 (x1 ), φ0τ 0 (h1 (x2 ))) < ε
(9)
and for any x01 , x02 ∈ Iφ0 (V 0 ) where d(x01 , x02 ) < δ there exists |τ | < τ0 such that d(h01 (x1 ), φτ (h01 (x2 )))) < ε.
(10) Lastly, we know (11)
h1 (Iφ (V )) ⊂ Iφ0 (U 0 ), h01 (Iφ0 (V 0 )) ⊂ Iφ (U ),
(12)
Iφ0 (V 0 ) ⊂ h1 (Iφ (V )), Iφ (V ) ⊂ h01 (Iφ0 (V 0 )),
and for any x ∈ Iφ (U ) and x0 ∈ Iφ0 (U 0 ) we have (13)
h1 (O(x)) ⊂ O0 (h1 (x)),
(14)
h01 (O0 (x0 )) ⊂ O(h01 (x0 )),
and there exists |τ | < τ0 and |τ 0 | < τ00 such that (15)
h01 ◦ h1 (x) = φ(τ, x),
h1 ◦ h01 (x0 ) = φ0 (τ 0 , x0 ).
Remark 4.5. Note that (9), (10) are analogs of continuity for the maps h1 , h01 . Relation (15) state that h01 is almost an inverse of h1 . The reason we do not obtain continuous invertible maps is due to the nonuniqueness of shadowing for flows. Proof. Let X be the vector field generating the smooth flow φ. For a point x ∈ M and ε > 0 such that X(x) 6= 0 denote [ L(x, ε) := expx (v). v∈Tx M, |v| 0 and a neighborhood O of Y such that O ⊂ ∪x∈Y L(x, ε), and L(x, ε) ∩ L(φ(t, x), ε) = ∅ for x ∈ M, t ∈ (−τ1 , τ1 ). Let a, a0 , τ0 , τ00 > 0 be constants from the expansivity property for Λ and Λ0 .
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For δ > 0 denote by P (δ) the set of δ-pseudo orbits g(t) contained in Λ. For any τ ∈ R consider the mapping στ : P (δ) → P (δ) defined by the relation (στ g)(t) = g(t + τ ). Shadowing and expansivity imply that for any ε > 0 there exists δ > 0 such that for any g ∈ P (δ) there exists a unique point x ∈ L(g(0), ε) and a (not necessarily unique) reparametrization γ ∈ Rep such that (16)
d(g(t), φ(γ(t), x)) < ε.
Consider ε = a/2 and the corresponding δ0 . Now define a map T : P (δ0 ) → M such that x = T (g) is the unique point satisfying (16). For η > 0 we say that pseudo orbits g1 , g2 ∈ P (δ) are η-rep-close if there exists a reparametrization γ ∈ Rep such that d(g1 (t), g2 (γ(t))) < η. We will use the following properties of the map T . By shadowing and expansivity we know that for sufficiently small δ that for any g ∈ P (δ) and t ∈ R there exists t0 ∈ R such that T ◦ σt (g) = φt0 ◦ T (g). Fix small enough η, δ > 0 such that if g1 , g2 ∈ P (δ) are η-rep-close then there exists |τ | < τ0 such that (17)
T g2 = φ(τ, T g1 ).
Fix δ1 > 0 sufficiently small such that V0 = B(δ1 , Λ) ⊂ U . For any point z ∈ Iφ (V0 ) and t ∈ R let us choose a point g(t) ∈ Λ such that the inclusion φt (z) ∈ L(δ1 , g(t)) holds. We also assume that δ1 is sufficiently small so that the map g(t) is a δ-pseudo orbit. Define a map S : V0 → P (δ) as S(z) := g. We would like to emphasize that the choice of g(t) is not unique, however for any such choice T (g) = z. Again from the expansivity property we have T ◦ S = id. Also, for ε > 0 sufficiently small there exists Θ > 0 such that if g1 , g2 ∈ P (δ) and d(g1 (t), g2 (t)) < δ, |t| < Θ then there exists |τ | < τ0 d(S(g1 ), φ(τ, S(g2 ))) < ε. For δ possibly smaller we can fix η > 0 sufficiently small so that for any g ∈ P (δ) pseudo orbits g and S ◦ T g are η-rep-close. Additionally,
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T. FISHER, T. PETTY, AND S. TIKHOMIROV
there exists a neighborhood V1 ⊂ V0 of Λ such that for any x ∈ Iφ (V1 ) the inclusion S(x) ∈ P (δ) holds. Similarly, fix δ00 , δ10 , δ 0 , η 0 > 0, maps T 0 : P (δ00 ) → M 0 and S 0 : B(δ10 , Λ0 ) → P (δ 0 ), and a neighborhood V10 of Λ0 . Now we are ready to construct the maps h1 , h01 . Let us choose δ2 ∈ (0, δ) such that for any g ∈ P (δ2 ) the inclusion h(g) ∈ P (δ 0 ) holds. Now let us choose V ⊂ V1 an open neighborhood of Λ such that for any x ∈ Iφ (V ) the inclusion S(x) ∈ P (δ2 ) holds. Notice that if g is a pseudo-orbit contained strictly in Λ, then we can define a pseudoorbit in Λ0 by h(g). We now define the map h1 : Iφ (V ) → M 0 as h1 := T 0 ◦ h ◦ S. Similarly, define V 0 and map h01 : IX 0 (V 0 ) → M as h01 = T ◦ h−1 ◦ S 0 . Among the properties of maps h1 , h01 the most difficult is relation (15). We will give its proof in full details. Properties (13) and (14) will follow directly from the definitions of the functions h1 and h01 . Relations (8)–(12) can be easily deduced (decreasing V and V 0 if necessarily) from • properties described above, • expansivity of the vector fields in V, V 0 , and • continuity of h, h−1 . We prove only the second equality in (15). Note that h01 ◦ h1 = T ◦ h−1 ◦ S 0 ◦ T 0 ◦ h ◦ S. For η 0 > 0 perhaps smaller we know that if g10 and g20 are η 0 -rep-close then h−1 g10 , h−1 g20 are η-rep-close. Also, for δ 0 > 0 perhaps smaller we know that if g10 ∈ P (δ 0 ), then pseudo orbits g10 and S 0 ◦ T 0 g10 are η 0 -rep-close. By the continuity of h for V , perhaps smaller, we know that for any x ∈ Iφ (V ) the inclusion h ◦ Sx ∈ P (δ 0 ) holds. Let x ∈ V . Set g1 := Sx, g10 := hg1 , x0 := T 0 g10 , g20 := S 0 x0 , g2 := h−1 g20 , y := T g2 . By construction of V we know g10 ∈ P (δ 0 ). Note that g20 = S 0 ◦ T 0 g10 . Hence, g10 and g20 are η 0 -rep-close. Also, we know that g1 = h−1 g10 , and g2 = h−1 g20 . Hence, g1 and g2 are η-rep-close. Finally, we have x = T g1 ,
y = T g2 .
Now we know that (17) implies the second equality (15).
Proof of Theorem 1.6. We prove that if Λ is premaximal then Λ0 is premaximal. The converse statement can be proven similarly. Consider neighborhoods V ⊂ U of Λ and V 0 ⊂ U 0 of Λ0 and maps h1 , h01 from Theorem 4.4.
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Assume that Σ ⊂ V is a locally maximal set with isolating neighborhood W = B(η, Σ), such that Σ0 = O(h1 (Σ)) ⊂ V 0 . Below we will prove that Σ0 is locally maximal. Consider η1 > 0 such that for x ∈ B(η1 , Σ) and |τ | < τ0 the inclusion φτ (x) holds. Using equality (14) and inequality (10) for ε = η1 we find η10 > 0 such that if x0 ∈ W 0 := B(η10 , Σ0 ) then h01 (x0 ) ∈ B(η1 , Σ). Let us prove that Σ0 = Iφ0 W 0 . Assume contrarily that there exists 0 x ∈ W 0 \ Σ0 such that O(x0 ) ⊂ W 0 . Relations (13), (14), (15) imply that [ (18) O(h(x0 )) = φτ (h01 (x01 )). |τ | 0 any δ-pseudo orbit consisting of points from Λ can be shadowed by a trajectory from Λ for small enough δ). It easily leads us to the following. Proposition 4.6. If sh(Λ, δ) = Λ for some δ > 0. Then Λ is locally maximal. As a reminder if Λ is a locally maximal set, then for ε > 0 sufficiently small we know that Bε (Λ) is an isolating set for Λ. In [9] Crovisier provides an example of a hyperbolic set that is never included in a locally maximal one, and this example shows that to establish premaximality it is not enough to say that a hyperbolic set is included in a locally maximal one or that the shadowing closure stabilizes for some δ > 0.
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T. FISHER, T. PETTY, AND S. TIKHOMIROV
To be more precise, let Let A and B be 2 × 2 hyperbolic toral automorphisms such that A has two fixed points p and q and the hyperbolicity in A dominates the hyperbolicity in B. Let F be a diffeomorphism on the 4-torus defined by F0 (x, y) = (Ax, By) and fix r a fixed point of B. Crovisier proves that if we let V be a sufficiently small neighborhood of (q, r), then the set \ Λ= f n (T4 − V ) n∈Z
is a hyperbolic set and the only locally maximal hyperbolic set that Λ can be included in is the entire manifold T4 ; so this example is not premaximal. However, Λ is included in a locally maximal hyperbolic set and for δ > 0 sufficiently large may be in a shadowing closure that stabilizes. Proof of Theorem 1.7. We first consider the case of diffeomorphisms. Suppose that Λ is a hyperbolic set for a diffeomorphism f : M → M and suppose that for any neighborhood U of Λ there exists a δ > 0 such that the shadowing closure of Λ stabilizes inside U . Now the previous proposition shows that the stabilizer is a locally maximal set and Λ is premaximal. To prove the other implication let f : M → M be a diffeomorphism and let Λ be a premaximal set for f and V be a neighborhood of Λ. Then there exists some η > 0 such that [ Bη (Λ) = Bη (x) ⊂ V x∈Λ
and Λη = If (Bη (Λ)) is hyperbolic. ˜ be a locally maximal hyperbolic set such that Let Λ ˜ ⊂ Λη ⊂ Bη (Λ) ⊂ V. Λ⊂Λ ˜ is an isolating neighborhood of Λ ˜ and Fix ε ∈ (0, η) such that Vε (Λ) ˜ is ε shadowed in Λ ˜ fix δ ∈ (0, ε/2) such that every δ-pseudo orbit in Λ ˜ by a unique point in Λ. From the choice of constants above we know that Λ ⊂ sh(Λ, δ) ⊂ Λε . Furthermore, we know that each δ-pseudo orbit of Λ is a δ-pseudo orbit ˜ so there exists a unique point y ∈ Λ ˜ that is a ε-shadowing point of Λ ˜ Let Λ1 = sh(Λ, δ). of the pseudo orbit. Hence, sh(Λ, δ) ⊂ Λ. If Λ1 = Λ we know from the previous proposition that Λ is locally maximal. So suppose that Λ1 6= Λ and let ν1 = dH (Λ1 , Λ) where dH (·, ·) is the Hausdorff distance between the sets. More generally, let Λj+1 = sh(Λj , δ), νj+1 = dH (Λj+1 , Λj ) for j ∈ N. By the shadowing estimates we know that νj ∈ (0, ε).
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Claim 4.7. There exists γ > 0 such that for all j ∈ N if Λj 6= Λj+1 and Λj+1 6= Λj+2 , then either νj ≥ γ or νj+1 ≥ γ. Proof of Claim. Consider γ ∈ (0, δ/4) such that for all x, y ∈ M if d(x, y) < γ, then d(f (x), f (y)) < δ/4 and d(f −1 (x), f −1 (y)) < δ/4. Suppose that Λj 6= Λj+1 and Λj+1 6= Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . We will construct a δ-pseudo orbit in Λj that y ε-shadows. This will show that y ∈ Λj+1 , a contradiction. Then there exists some y0 ∈ Λj+1 and x0 ∈ Λj such that d(y, y0 ) < νj+1 and d(y0 , x0 ) < νj < γ. Then d(f (y), f (y0 )) < δ/4 and d(f (y0 ), f (x0 )) < δ/4. Also, we know there exists points y1 ∈ Λj+1 and x1 ∈ Λj such that d(f (y), y1 ) < νj+2 and d(y1 , x1 ) < νj+1 . Hence, d(f (x0 ), x1 ) ≤ d(f (x0 ), f (y0 )) + d(f (y0 ), f (y)) + d(f (y), y1 ) + d(y1 , x1 ) ≤ δ/4 + δ/4 + νj+2 + νj+1 < δ and d(f (y), x1 ) ≤ d(f (y), y1 ) + d(y1 , x1 ) < νj+2 + νj+1 < δ/2 < ε. Continue inductively, we can construct a forward δ-pseudo orbit (xk )∞ k=0 such that y ε-shadows the pseudo orbit. Also, since the estimates on γ apply for f −1 we can construct a bi-infinite δ-pseudo orbit (xk ) in Λj such that y ε shadows (xk ). Then y ∈ Λj+1 , a contradiction. We now return to the proof of the theorem. ˜ for all Repeating the above arguments for Λj we see that Λj ⊂ Λ j ∈ N. We know from Proposition 4.6 that if Λj+1 = Λj for some j ∈ N that Λj is locally maximal. To conclude the proof of the theorem we simply need to show that the sequence Λj stabilizes. Suppose that the sequence Λj does not stabilizer. We know from the above claim that for each j the sets Λj+1 or Λj+2 will be a distance of γ from the set Λj using the Hausdorff metric. Inside a compact metric space we know an increasing sequence of compact sets converges in the Hausdorff topology. Hence, if the sequence does not stabilize there exists some N ∈ N where the Hausdorff distance from ΛN to Λ is greater ˜ and Λ ˜ ⊂ Vη (Λ). than η. This is a contradiction since each Λj ⊂ Λ Theorem 1.7 is proved for the case of diffeomorphisms. The proof for the case of flows follows the same ideas. Below we provide a detailed proof of the Claim 4.7, which is the central step of the proof of Theorem 1.7.
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We will use the following construction. For a sequence of points {xk }k∈Z consider a map g{xk } : R → M defined as g{xk } (t) := φ(t − [t], x[t] ), where [t] is the integer part of t. Consider map r : M → M defined as r(x) = φ(1, x). Note that r does not necessarily satisfy the shadowing property. We will use the following. F1: There exists ε1 > 0 such that for any sequence {xk } and point y satisfying (19)
d(xk , f k (y)) < ε1 ,
k∈Z
the following inequality holds d(g{xk } (t), φ(t, y)) < ε. Without loss of generality we can assume that δ < ε1 . F2: There exists δ1 > 0 such that if {xk } with xk ∈ Λj is a δ1 -pseudo orbit of the map r then g{xk } ⊂ Λj is a δ-pseudo orbit of the flow φ. Consider γ ∈ (0, δ1 /4) such that for all x, y ∈ M if d(x, y) < γ, then d(r(x), r(y)) < δ1 /4 and d(r−1 (x), r−1 (y)) < δ1 /4. As in the case of diffeomorphisms suppose that Λj 6= Λj+1 and Λj+1 6= Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . Arguing similarly to the case of diffeomorphisms we can construct a δ1 -pseudo orbit {xk } ⊂ Λj of the map r which satisfies (19). Properties F1 and F2 implies that δ-pseudo orbit g{xk } is ε-shadowed by y. This shows that y ∈ Λj+1 , a contradiction. References [1] V. M. Alekseev. Symbolic dynamics. Eleventh Mathematical School (Summer School, Kolomyya, 1973), pp. 5-210. Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1976. [2] D. V. Anosov. Intrinsic character of one property of hyperbolic sets. J. Dyn. Control Syst. 16, no. 4, 485-493, 2010. [3] D. V. Anosov. Extension of zero-dimensional hyperbolic sets to locally maximal ones. Math. Sb. 201, no. 7, 3-14, 2010. [4] D. V. Anosov. On certain hyperbolic sets. Mat. Zametki, 87, no. 5, 650-668, 2010. [5] D. V. Anosov. Local maximality of hyperbolic sets. Proc. Steklov Inst. Math. 273, no. 1, 23-24, 2011. [6] D. V. Anosov. On trajectories located entirely near a hyperbolic set. Sovrem. Mat. Fundam. Napravl. 45, 5–17, 2012.
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[7] V. Ara´ ujo, V. and M. J. Pac´ıfico, Three dimensional flows, Publica¸c˜oes Matem´ aticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA), Rio de Janeiro, 2007. [8] K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), no. 5, 1841-1872. [9] S. Crovisier. Une remarque sur les ensembles hyperboliques localement maximaux. C. R. Math. Acad. Sci. Paris, 334, no. 5, 401-404, 2001. [10] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226. [11] N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137. [12] N. Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), no. 1, 81-93. [13] T. Fisher. Hyperbolic sets that are not locally maximal. Ergod. Th. Dynamic. Systems, 26, no. 5, 1491-1509, 2006. [14] T. Hunt, Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity, PhD Thesis, University of Cambridge, Queen’s College, 2000. [15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995). [16] Y. Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, 16 Birkh¨ auser Boston, Inc., Boston, MA, 1988. [17] T. Petty, Nonlocally maximal hyperbolic sets for flows, Maters Thesis, Brigham Young University, 2015. [18] S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer, Berlin, 1999. T. Fisher, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected] T. Petty, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected] S. Tikhomirov, Max Planck Institute for Mathematics in the sciences, Inselstrasse 22, 04103, Leipzig, Germany; Chebyshev Laboratory, Saint-Petersburg State University, 14th Line 29B, Vasilyevsky Island, St.Petersburg 199178, Russia E-mail address: [email protected]
1. Introduction Since the 1960s the study of hyperbolic sets has been a cornerstone in the field of dynamical systems. These sets are remarkable not only in their complexity, but also in the fact that they persist under perturbations. Additionally, for a point in a hyperbolic set the derivative of the map at this point gives information on the local dynamics for the original nonlinear map. As a reminder, for a diffeomorphism f : M → M , a compact invariant set Λ is hyperbolic for f if TΛ M = Es ⊕ Eu is a Df -invariant splitting such that Es is uniformly contracted and Eu is uniformly expanded by Df . Anosov was one of the pioneers in studying hyperbolic sets. Indeed, if the entire manifold is a hyperbolic set for a diffeomorphism, then the diffeomorphism is called Anosov. This is one of the best understood classes of hyperbolic sets. 2000 Mathematics Subject Classification. 37D20, 37D05, 37C05. Key words and phrases. hyperbolic sets, hyperbolic flow, premaximal, locally maximal, isolated. The authors would like to thank the Sixth International Conference on Differential and Functional Differential Equations during which much of the paper was prepared. T.F. is supported by Simons Foundation grant # 239708. S. T. is partially supported by Chebyshev Laboratory under Russian Federation Government grant 11.G34.31.0026, JSC “Gazprom neft”, Saint Petersburg State University research grant 6.38.223.2014, Russian Foundation of Basic Research 15-01-03797a and German-Russian Interdisciplinary Science Center (G-RISC). 1
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Another important class of hyperbolic sets are those that are locally maximal. For a compact metric space X and a continuous homeomorphism T : X → X a set K ⊂ X is locally maximal if there exists a neighborhood U of K such that \ K= T n (U ). n∈Z
Hence, such sets are the maximal invariant set within U . Locally maximal sets were defined more or less simultaneously by Anosov and Conley (who called these sets isolated) in the 1960s. Alexseev proved in [1] that a shift space is locally maximal if and only if it is a shift of finite type. Furthermore, he proved that given any shift space Σ and a neighborhood U of Σ there exists a locally maximal shift Σ0 containing Σ and contained in U . A question that was posed in the 1960s by Anosov, Alexseev, and others (that is stated for instance in [15, p. 272]), is the following: Question 1.1. If Λ is a hyperbolic set and U is a neighborhood of Λ, ˜ such that Λ ⊂ Λ ˜ ⊂ U? then is there a locally maximal hyperbolic set Λ As stated by Anosov in [3] it was hoped at the time that the answer would be in the affirmative. One reason this was hoped for is that locally maximal hyperbolic sets are more easily classified. Indeed, a hyperbolic set is known to be locally maximal if and only if it has a local product structure (defined in Section 2). A standard assumption used in characterizing topological and/or measure theoretic properties of hyperbolic sets is that the set is locally maximal. For instance, Smale’s Spectral Decomposition Theorem, see for instance [15, p. 575], is valid for locally maximal hyperbolic sets. In fact, quite frequently even when this is not specifically stated in a theorem one finds in the proof that this assumption is required for the result to hold. 1.1. Nonlocally maximal hyperbolic sets. It was shown by Crovisier in [9] that there is a hyperbolic set on the 4-torus that is never included in a locally maximal set. Later, in [13] it was shown that any compact boundaryless manifold with dimension greater than or equal to 2 has a C r open set of diffeomorphisms where 1 ≤ r ≤ ∞ such that each diffeomorphism in the open set contains a hyperbolic set that is not included in a locally maximal one. Our first result is an extension of the results in [13] to hyperbolic flows. As a reminder, for a smooth flow φ : R × M → M , a compact φ-invariant set Λ is hyperbolic for φ if TΛ M = Es ⊕ Ec ⊕ Eu is a flow invariant splitting such that Es is uniformly contracted, Ec is the flow
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direction, and Eu is uniformly expanded. A set Λ is locally maximal for the flow φ if there is an open set U containing Λ such that Λ = ∩t∈R φt (U ). Theorem 1.2. Let M be a compact, boundaryless C r manifold for 1 ≤ r ≤ ∞ with dim M ≥ 3 and X k (M ) be the set of C k flows on M where 1 ≤ k ≤ r. Then there exists a C k open set of flows on M such that each flow contains a hyperbolic set not contained in a locally maximal one. Remark 1.3. We notice the following. 1. If dim M ≤ 2 then every hyperbolic set for a smooth flow is a finite union of hyperbolic closed trajectories and hence it is locally maximal. 2. Also, one can always suspend the map constructed in [13] and have a topological flow. However, it remains to be seen if the suspension will still be smooth. Also, the suspension could introduce nontrivial topology and it may not be possible to obtain the result of 1.2 on any manifold of dimension 3 or larger. As in [13, Theorem 1.5] we can show the following result. Theorem 1.4. Let Λ be a hyperbolic set for a flow and U be a neighborhood of Λ, then there exists a hyperbolic set Λ0 with a Markov partition for the flow such that Λ ⊂ Λ0 ⊂ U . The proof is very similar to that in [13]. Indeed, the necessary theorems used in [13] to prove the similar results for maps hold for hyperbolic sets for flows, and hence the proof is left to the reader. The hyperbolic set constructed in Theorem 1.2 above need not transitive under the flow. However, as in [13, Theorem 1.6] one can construct a flow in higher dimensions with a transitive hyperbolic set that is not contained in a locally maximal one. 1.2. Premaximality. In this paper we also examine conditions under which a hyperbolic set, Λ, is included in a locally maximal hyperbolic set within an arbitrarily small neighborhood of Λ. Following the terminology introduced by Anosov in [3] we define a hyperbolic set Λ for a diffeomorphism to be premaximal if for any open ˜ such set U containing Λ there is a locally maximal hyperbolic set Λ ˜ ⊂ U . In [3] Anosov proves that any zero-dimensional that Λ ⊂ Λ hyperbolic set for a diffeomorphism is premaximal, and in [2] Anosov proves there is an intrinsic property for premaximal hyperbolic sets for diffeomorphisms. Moreover the following holds.
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Theorem 1.5. [6] Let f : M → M and f 0 : M 0 → M 0 be diffeomorphisms, Λ a hyperbolic set for f , Λ0 a hyperbolic set for f 0 , and h : Λ → Λ0 a homeomorphism such that h ◦ f = f 0 ◦ h. If U is a neighborhood of Λ and U 0 is a neighborhood of Λ0 , then there exists neighborhoods V ⊂ U of Λ and V 0 ⊂ U 0 of Λ0 and continuous injective equivariant maps h1 : If (U ) → M 0 and h2 : If 0 (U 0 ) → M such that h1 |Λ = h, and h1 (If (V )) ⊂ If 0 (U 0 ),
h2 (If 0 (V 0 )) ⊂ If (U ), f ◦ h2 |If 0 (V 0 ) = h2 ◦ f 0 |If 0 (V 0 ) ,
h1 ◦ f |If (V ) = g ◦ h1 |If (V ) , h2 ◦ h1 |If (V ) = id, and
h1 ◦ h2 |If 0 (V 0 ) = id.
The above theorem shows that f |Λ defines the set of trajectories that lie in a sufficiently small neighborhood of Λ. However, in [6] the specific intrinsic property for premaximality is not stated. We extend result of [6] to the case of flows and prove the premaximality is an intrinsic property for hyperbolic sets for flows. Let X and X 0 be vector fields on smooth compact Riemannian manifolds M and M 0 respectively. Denote by φ and φ0 flows generated by them. An increasing homeomorphism of the real line α : R → R is called a reparametrization. Let Λ and Λ0 be hyperbolic sets for X and X 0 respectively. We say that Λ and Λ0 are topologically equivalent if there exists a homeomorphism h : Λ → Λ0 and a continuous map α : M × R → R such that h ◦ φt (x) = φ0 (α(x, t), h(x)),
x ∈ Λ, t ∈ R
where α(x, ·) is a reparametrization for each x ∈ Λ. In this case there exists a continuous map β : M 0 × R → R, such that β(h(x), α(x, t)) = t,
x ∈ Λ, t ∈ R
α(h−1 (x0 ), β(x0 , t0 )) = t0 , and x0 ∈ Λ0 , t0 ∈ R. Theorem 1.6. Let Λ and Λ0 be hyperbolic sets for vector fields X and X 0 respectively. Assume that Λ and Λ0 are topologically equivalent. Then Λ is premaximal if and only if Λ0 is premaximal. Below we provide an equivalent condition to premaximality for diffeomorphisms and flows. Before stating the result we define some important terms involving shadowing. Let Φ = φ(t, x) be a dynamical system where t can be taken to be discrete or continuous. If t is discrete, we assume the dynamical system is generated by a diffeomorphism of a compact manifold to itself. If t is continuous, we assume that the dynamical system is generated by a smooth vector field on a compact
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manifold. Let a > 0 be an expansivity constant for some neighborhood of a hyperbolic set Λ and let δ0 > 0 be such that any δ0 -pseudo orbit in Λ can be a/2-shadowed by an exact trajectory (definitions are given in the next section). Note that due to expansivity the shadowing trajectory is unique (for the case of a flow this is true up to a reparametrization). For Λ a hyperbolic set for a diffeomorphism and δ ∈ (0, δ0 ) the shadowing closure (or δ-shadowing closure) of Λ is sh(Λ, δ) = {y ∈ M : y shadows a δ-pseudo orbit in Λ}. For a fixed δ > 0 we can construct a sequence of shadowing closures Λ0 , Λ1 , ..., where Λ0 = Λ and Λj = sh(Λj−1 , δ) for j ∈ N. We say a shadowing sequence stabilizes if Λj = Λj+1 for all j ≥ N where N ∈ N. Theorem 1.7. For a hyperbolic set Λ of a dynamical system Φ the following statements are equivalent (1) Λ is premaximal; (2) for any neighborhood U of Λ the shadowing closure stabilizes inside U for some δ > 0. Note that due to Theorem 1.5 and Theorem 1.6 the second property in Theorem 1.7 for diffeomorphisms is intrinsic. The paper proceeds as follows. In Section 2 we review relevant background on hyperbolicity and flows. In Section 3 we prove Theorem 1.2. In Section 4 we review the results of Anosov in [2, 3, 4, 5, 6] and prove Theorems 1.6 and 1.7. 2. Background 2.1. Hyperbolic sets for flows. We first review properties of hyperbolic sets for flows. Definition 2.1. Let X be a metric space and φ a continuous flow on X. Then for x ∈ X we define the stable set W s (x) := {y ∈ X : lim d(φt (x), φt (y)) = 0}. t→∞
Further, for ε > 0 the ε-stable set is Wεs (x) := {y ∈ W s (x) : d(φt (x), φt (y)) ≤ ε for all t ≥ 0}. Note that the unstable sets W u (x) and Wεu (x) are defined identically under the flow φ−t . Furthermore, we define the center-stable set [ W cs (x) := {φt (W s (x))|t∈R } = W s (y). y∈φt (x) t∈R
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The center-unstable set of x is defined to be the center-stable set of x s under φ−t . We will also use the notation Wloc to mean Wεs for sufficiently u similarly to mean Wεu for small ε (depending on the context) and Wloc small ε. Let X be a manifold and φ be a C r flow on X. If Λ is a hyperbolic set for φ and p ∈ Λ, then W s (p) is a C r immersed submanifold and is an immersed copy of Rk where k = dim Es (x). Similar statements hold for the unstable sets, center-stable sets, and center-unstable sets. Also note that the stable and unstable manifolds vary continuously on the point p. Definition 2.2. For a metric space X and a flow φ, a set Γ ⊂ X is said to have a local product structure if for all ε > 0 sufficiently small there exists a δ > 0 such that given x, y ∈ Γ with d(x, y) < δ we have, for some real |t| < ε, a unique point S(x, y) := b ∈ Wεu (φt (x)) ∩ Wεs (y) contained in Γ. Remark 2.3. Note that for any hyperbolic set Λ there always exist constants δ and ε sufficiently small such that x, y ∈ Λ and d(x, y) < δ implies S(x, y) = Wεu (φt (x)) ∩ Wεs (y) is a unique point in the manifold, but may not be in Λ. The following lemma is also critical to the paper. Note that this lemma is almost always stated and proved for maps, but is in fact true for flows as well (see [8] and [16]). Lemma 2.4. A hyperbolic set Γ has a local product structure if and only if it is locally maximal. We also need the notion of the shadowing property. For δ > 0 a map g : R → M is an δ-pseudo orbit if the following holds d(g(t + τ ), φτ (g(t))) < δ,
t ∈ R, |τ | < 1.
A δ-pseudo orbit g is ε-shadowed by a point x0 if there exists a reparametrization α : R → R satisfying d(g(t), φα(t) (x0 )) < ε, and α(t1 ) − α(t2 ) < ε for t1 6= t2 . − 1 t1 − t2 A vector field X is expansive on a compact metric space W if there exist constants a, τ0 > 0 such that if x1 , x2 ∈ W and there exists a reparametrization α : R → R such that the following inequalities hold d(φ(α(t), x1 ), φ(t, x2 )) < a, then x2 = φ(τ, x1 ), where τ ∈ (−τ0 , τ0 ).
t ∈ R,
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Theorem 2.5. Let Λ be a hyperbolic set for a flow φ on a compact manifold M . Then the following hold: • there exists a neighborhood W of Λ such that φ is expansive on \ Iφ (W ) := φt (W ); t∈R
and • there exists a neighborhood U (Λ) ⊂ W such that for any ε > 0 there exists δ > 0 such that any δ-pseudo orbit g ⊂ U can be ε-shadowed by some point x0 ∈ M . Definition 2.6. For φ a flow on a compact metric space X a nonempty subset A of X is called an attractor if it satisfies the following three conditions. (i) A is forward-invariant under φ; i.e., x ∈ A implies φt (x) ∈ A for all t > 0. (ii) There exists a neighborhood of A, called the basin of attraction of A and denoted B(A), which consists of all points that tend towards A under φt as t → ∞. In other words, B(A) consists of all points x such that for any open neighborhood N of A, there exists T > 0 such that φt (x) ∈ N for all t > T . (iii) No proper subset of A satisfies conditions (i) and (ii). When an attractor Λ is (uniformly) hyperbolic we know that • periodic points are dense in Λ, • x ∈ Λ implies W cu (x) ⊂ Λ, and S • for any periodic point x ∈ Λ we know y∈O(x) W cs (y) is dense in B(Λ). We will need the following technical result, known in the literature as the Inclination Lemma, or λ-lemma. The statement can be found in [7]. Note that the statement for hyperbolic periodic points would be similar. Lemma 2.7 (Inclination Lemma). Let p ∈ M be a hyperbolic fixed point for a C r flow φ, for r ≥ 1, with local stable and unstable manifolds s u u Wloc (p) and Wloc (p), respectively. Fix an embedded disk B in Wloc (p) u which is a neighborhood of p in Wloc (p), and fix a neighborhood V of s this disk in M. Let D be a transverse disk to Wloc (p) at a point z such that D and B have the same dimension. Write Dt for the connected component of φt (D) ∩ V which contains φt (z), for t ≥ 0. Then, given ε > 0 there exists T > 0 such that for all t > T the disk Dt is ε-close to B in the C r -topology.
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2.2. Hyperbolic sets of diffeomorphisms. Since the statement of Theorem 1.7 refers to diffeomorphisms as well as smooth flows we now review some definitions for discrete dynamical systems. For f : X → X a homeomorphism of a metric space and δ > 0 an δ-pseudo orbit is a sequence {xj }m l where • l ∈ {−∞} ∪ Z, m ∈ Z ∪ {∞}, l < m, and • d(f (xj ), xj+1 ) < δ for all j ∈ [l, m]. For a δ-pseudo orbit {xj }m l we say this sequence is ε-shadowed by a j point x ∈ X if d(f (x), xj ) < ε for j ∈ [l, m]. We say that a homeomorphism f of a compact metric space W is expansive if there exists a constant a > 0 such that if \ f n (W ) x1 , x2 ∈ If (W ) := n∈Z
and d(f n (x1 ), f n (x2 )) < a,
n ∈ R,
then x2 = x1 . Theorem 2.8. (Shadowing Lemma) Let Λ be a hyperbolic set for f : M → M a diffeomorphism. Then there exists neighborhood U (Λ) such that for all ε > 0 there exists an δ > 0 such that if {xj }∞ −∞ ⊂ U is an δ-pseudo orbit, then there exists x ∈ M that ε-shadows {xj }∞ −∞ . Let f : M → M be a diffeomorphism and Λ be a hyperbolic set for f . For ε > 0 sufficiently small and x ∈ Λ the local stable and unstable manifolds and stable and unstable manifolds are defined similar to the case for flows and for a C r diffeomorphism f the stable and unstable manifolds of a hyperbolic set are C r injectively immersed submanifolds. For Λ a hyperbolic set we know that if ε is sufficiently small and x, y ∈ Λ, then Wεs (x) ∩ Wεu (y) consists of at most one point. For such an ε > 0 define Dε = {(x, y) ∈ Λ × Λ | Wεs (x) ∩ Wεu (y) ∈ Λ} and [·, ·] : Dε → Λ so that [x, y] = Wεs (x) ∩ Wεu (y). We will also need openness of hyperbolicity. Lemma 2.9. Let Λ ⊂ M be a hyperbolic set of the diffeomorphism f : U → M . Then for any open neighborhood V ⊂ U of Λ and every δ > 0 there exists ε > 0 such that if f 0 : U → M and dC 1 (f |V , f 0 ) < ε there is a hyperbolic set Λ0 = f 0 (Λ0 ) ⊂ V for f 0 and a homeomorphism h : Λ0 → Λ with dC 0 (Id, h)+dC 0 (Id, h−1 ) < δ such that h◦f 0 |Λ0 = f |Λ ◦h. Moreover, h is unique when δ is sufficiently small.
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2.3. Normal hyperbolicity. For embedding constructions into higher dimensions, we will need the notion of normal hyperbolicity. A normally hyperbolic invariant manifold (NHIM) is a generalization of a hyperbolic fixed point and a hyperbolic set. Fenichel proved that NHIMs and their stable and unstable manifolds are persistent under perturbation [11], [12]. We define NHIMs for maps, but the definition for flows is similar (and more technical). Definition 2.10. Let M be a compact smooth manifold and f : M → M a diffeomorphism. Then an f -invariant submanifold Λ of M is said to be a normally hyperbolic invariant manifold if there exist m ∈ N such that the mapping f m satisfies the following property. There exists a continuous invariant bundle TΛ M = E s ⊕ T Λ ⊕ E u (x), and continuous positive functions ν, νˆ, γ, γˆ : M → R such that ν, νˆ < 1,
ν < γ < γˆ < νˆ−1
and for all x ∈ Λ, v ∈ Tx M , |v| = 1 |Df m (x)v| ≤ ν(x), v ∈ E s (x); γ(x) ≤ |Df m (x)v| ≤ γˆ (x), v ∈ T Λ; |Df m (x)v| ≥ νˆ−1 (x), v ∈ E u (x). Adapting the above for flows gives us an important result ([10, p. 215]) which says that if a C r vector field Y in some C 1 neighborhood of our original vector field X (equated with a flow φ, under which Λ is invariant) there is a C r manifold ΛY invariant under Y and C r diffeomorphic to Λ. An immediate consequence of this is that the dynamics on ΛY under the vector field Y are a perturbation of the dynamics of Λ under X. 3. Nonlocally maximal sets for flows The foundation of the proof of Theorem 1.2 is the Plykin attractor, see for instance [15, p. 537-41] for a construction of the Plykin attractor. The first author used this map with some modifications to prove Theorem 1.3 in [13] on the existence of hyperbolic sets not included in locally maximal ones. To extend the results of [13] to flows we need to show the suspension can be done smoothly and in such a way that no nontrivial topology is introduced in the suspension. In [14] Hunt shows that the Plykin attractor can be suspended smoothly in such a way that the suspended flow is on the solid 2torus. The result of the construction is a smooth flow of a solid 2-torus where the basin of attraction for the suspended Plykin attractor is the
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interior of the torus. Using that flow, we apply the construction in [13] to Hunt’s flow and address the technicalities. Proof of Theorem 1.2. We now show how to adapt the construction in [13] to the suspension flow described above. First we would like to sketch the main steps of the construction of the example for a more detailed proof see [17]. Let T1 be the solid 2-torus used in the construction described above. We first embed Hunt’s Plykin attractor in a slightly larger solid 2-torus, T , so that the flow is the identity in a neighborhood of the boundary and then extend the flow to a 3-disk, D, so that it is the identity on D − T . This will allow us to embed the example into any arbitrary 3-manifold. We now modify the flow on T1 to incorporate the construction in [13]. Let Λa be the Plykin attractor and φ be the flow on D. Fix some p ∈ ∂T1 . Take a sufficiently small open neighborhood U of O(p), small enough to be disjoint from ∂T and Λa , and alter φ in U so that p is a hyperbolic saddle periodic point with W s (p) ⊂ U ∩ ∂T1 . Also, W cu (p) \ O(p) contains two components. One of which is contained in T − T1 and the other, denoted W ∗ (p) is contained strictly in the interior of T1 . Let q be a periodic point in Λa . Since W cs (q) = W s (Λa ) = int(T1 ) we know that given any point of W ∗ (p) that there must exist some point in W cs (q) arbitrarily close to it. Fix z ∈ W ∗ (p). Perturb the flow in a neighborhood of z so that z ∈ W cs (q) t W ∗ (p). This can be done since z is a wandering point for the flow. Here we will need two definitions. A hyperbolic set Λ for a C 1 flow has a heteroclinic tangency if there exist x, y ∈ Λ such that W s (x) ∩ W u (y) contains a point of tangency. A point of quadratic tangency for a C 2 flow is defined as a point of heteroclinic tangency where the curvature of the stable and unstable manifolds differs at the point of tangency. Now after a further perturbation to the flow as in [13] there exists a point w ∈ W u (z) and a point q 0 ∈ W u (q) such that W u (z) and W s (q 0 ) have a quadratic tangency at w. Let I be the segment of W u (p) from z to w, and let J be the segment of W u (q) from q to q 0 . The resulting flow will contain a hyperbolic set that cannot be contained in a locally maximal set as we show below. Figure 1 demonstrates a cross section of the constructed flow. Let Λ = Λa ∪ O(p) ∪ O(z). Standard arguments as in [13], adapted to flows, show that this is a hyperbolic set under φ. Now suppose
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w
z
q
J
q‘
Figure 1. Intervals I and J. Λ ⊂ Λ0 , where Λ0 is a locally maximal hyperbolic set. Modifying the proof in [13] we see from the product structure on Λ0 that this implies w ∈ Λ0 , a contradiction. Hence, Λ is not contained in a locally maximal hyperbolic set. We now show the construction is robust under perturbation. Since transversality is trivially open, and hyperbolicity is open by Lemma 2.9, it is sufficient to show that there remains a point w˜ ∈ W u (˜ p) ∩ W s (˜ u) cu for some u˜ ∈ Wloc (˜ q ). Let p˜ and q˜ be the continuations of p and q for the perturbed flow. By construction, the stable manifolds for all cu the x ∈ Wloc (˜ q ) locally foliate the region, so there must exist a point cu q ) and a point w˜ ∈ W cs (˜ u) ∩ W u (˜ p) such that the oneu˜ ∈ Wloc (˜ u dimensional path W (˜ p) remains tangent to the two-dimensional plane W cs (˜ u) at w. ˜ Specifically, we have Tw˜ W u (˜ p) ( Tw˜ W cs (˜ u). Using normal hyperbolicity we can embed our example into any smooth manifold M of dimension greater than 3. Furthermore, normal hyperbolicity implies the construction is still robust in this setting. 2 4. Premaximality Before we proceed to the proof of Theorems 1.6 and 1.7 let us review results by Anosov on premaximality [2, 3, 4, 5, 6]. As was mentioned in Section 1, Theorem 1.5 implies that premaximality is an intrinsic property.
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Let us recall the main idea of the proof of Theorem 1.5. For δ > 0 denote P (δ) as the set of δ-pseudo orbits {yn ∈ Λ}n∈Z endowed with the Tikhonov product topology. Consider the shift map σ : P (δ) → P (δ) defined as σ({yn }) = {yn+1 }. The Shadowing Lemma implies that for any ε > 0 there exists δ > 0 such that for any {yn } ∈ P (δ) there exists point x such that (1)
d(yn , f n (x)) < ε.
Note that due to the expansivity property we know for small enough ε > 0 that such a point x is unique. Fix such an ε > 0 and a corresponding δ from the shadowing lemma. Consider the map T : P (δ) → M defined by the condition that for {yn } ∈ P (δ) the point x = T ({yn }) is the unique point satisfying (1). It is easy to show that (2)
T ◦ σ = f ◦ T.
Now let us consider c > 0 and a small neighborhood U ⊂ B(c, Λ) of Λ and a point z such that O(z) ⊂ U . There exists a sequence of points {yn } satisfying (3)
d(yn , f n (z)) < c.
For any δ > 0 there exists a c > 0 such that inequality (3) implies {yn } ∈ P (δ) and (4)
T ({yn }) = z.
We can similarly define for Λ0 and f 0 sets P 0 (δ) and maps T 0 , σ 0 . Note that similarly to (2) the equality (5)
T 0 ◦ σ0 = f 0 ◦ T 0
holds. For any δ10 > 0 there exists δ1 > 0 such that (6)
h(P (δ1 )) ⊂ P 0 (δ10 ).
(Recall that h : Λ → Λ0 is a conjugacy between Λ and Λ0 .) Similarly for any δ2 there exists δ20 such that (7)
h−1 (P 0 (δ20 )) ⊂ P (δ2 ).
Equations (2), (4), (5), (6), and (7) allow us to conclude Theorem 1.5. For a detailed exposition of the proof we refer the reader to the original paper [6]. To study premaximality of the zero-dimensional hyperbolic sets we need the following notion. Let A be an “alphabet”: A = {1, . . . , n}
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and Ω = AZ equipped with the Tikhonov topology and metric X 1 d(a, b) = I(ai , bi ), |i| 2 i∈Z where a = (ai ), b = (bi ) ∈ Ω and I(ai , bi ) equal to 0 if ai = bi and 1 otherwise. Consider the shift map σ = Ω → Ω defined as (σ(a))i = ai+1 . Consider some set W of admissible words of lengths k ≥ 1 in the alphabet A and consider MW ⊂ Ω such that all subwords of all a ∈ MW of length k are admissible. Theorem 4.1. [1] Consider a set Λ ⊂ Ω. The following holds (1) the set Λ is locally maximal for σ if and only if there exists k ≥ 1 and set of admissible words W such that Λ = MW ; (2) the set Λ is premaximal. In [3] it was proved that zero-dimensional hyperbolic sets are topologically conjugated to Bernoulli shifts, which implies the next result. Theorem 4.2. Let Λ be a zero-dimensional hyperbolic set of a diffeomorphism f . Then Λ is premaximal. Burns and Gelfert were able to extend the above result to prove that a 1-dimensional hyperbolic set for a flow is premaximal [8, Proposition 8]. The proof follows an argument provided by Anosov after personal communications. In [4] Anosov obtain the following sufficient condition for a hyperbolic set to not be premaximal. Theorem 4.3. Let Λ be a hyperbolic set of f ∈ C 1 . Assume that there exists a family of exact trajectories ξ : Z × [0, a] → M such that (1) ξn+1,t = f (ξ(n, t)), (2) ξ(0, 0) ∈ Λ, (3) d(ξ(n, t), Λ) → 0 uniformly in t as |n| → ∞, and (4) there exists t1 ∈ [0, a] such that ξ(0, t1 ) ∈ / Λ. Then Λ is not premaximal. Note that the examples of Crovisier and Fisher satisfy these conditions. We now prove an analog of Theorem 1.5 for flows from which Theorem 1.6 will follow. Theorem 4.4. Let Λ and Λ0 be hyperbolic sets for smooth flows φ and φ0 respectively. Assume that Λ and Λ0 are topologically equivalent, with the corresponding map h : Λ → Λ0 . If U is a neighborhood of Λ and U 0 is a neighborhood of Λ0 , then there exists neighborhoods V ⊂ U of Λ
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and V 0 ⊂ U 0 of Λ0 , numbers τ0 , τ00 > 0, and (not necessarily continuous) maps h1 : Iφ (V ) → M 0 and h01 : Iφ0 (V 0 ) → M such that h1 |Λ = h and h01 |Λ0 = h−1 .
(8)
Furthermore, for any ε > 0 there exists δ > 0 such that if x1 , x2 ∈ Iφ (V ) and d(x1 , x2 ) < δ, then there exists |τ 0 | < τ00 such that d(h1 (x1 ), φ0τ 0 (h1 (x2 ))) < ε
(9)
and for any x01 , x02 ∈ Iφ0 (V 0 ) where d(x01 , x02 ) < δ there exists |τ | < τ0 such that d(h01 (x1 ), φτ (h01 (x2 )))) < ε.
(10) Lastly, we know (11)
h1 (Iφ (V )) ⊂ Iφ0 (U 0 ), h01 (Iφ0 (V 0 )) ⊂ Iφ (U ),
(12)
Iφ0 (V 0 ) ⊂ h1 (Iφ (V )), Iφ (V ) ⊂ h01 (Iφ0 (V 0 )),
and for any x ∈ Iφ (U ) and x0 ∈ Iφ0 (U 0 ) we have (13)
h1 (O(x)) ⊂ O0 (h1 (x)),
(14)
h01 (O0 (x0 )) ⊂ O(h01 (x0 )),
and there exists |τ | < τ0 and |τ 0 | < τ00 such that (15)
h01 ◦ h1 (x) = φ(τ, x),
h1 ◦ h01 (x0 ) = φ0 (τ 0 , x0 ).
Remark 4.5. Note that (9), (10) are analogs of continuity for the maps h1 , h01 . Relation (15) state that h01 is almost an inverse of h1 . The reason we do not obtain continuous invertible maps is due to the nonuniqueness of shadowing for flows. Proof. Let X be the vector field generating the smooth flow φ. For a point x ∈ M and ε > 0 such that X(x) 6= 0 denote [ L(x, ε) := expx (v). v∈Tx M, |v| 0 and a neighborhood O of Y such that O ⊂ ∪x∈Y L(x, ε), and L(x, ε) ∩ L(φ(t, x), ε) = ∅ for x ∈ M, t ∈ (−τ1 , τ1 ). Let a, a0 , τ0 , τ00 > 0 be constants from the expansivity property for Λ and Λ0 .
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For δ > 0 denote by P (δ) the set of δ-pseudo orbits g(t) contained in Λ. For any τ ∈ R consider the mapping στ : P (δ) → P (δ) defined by the relation (στ g)(t) = g(t + τ ). Shadowing and expansivity imply that for any ε > 0 there exists δ > 0 such that for any g ∈ P (δ) there exists a unique point x ∈ L(g(0), ε) and a (not necessarily unique) reparametrization γ ∈ Rep such that (16)
d(g(t), φ(γ(t), x)) < ε.
Consider ε = a/2 and the corresponding δ0 . Now define a map T : P (δ0 ) → M such that x = T (g) is the unique point satisfying (16). For η > 0 we say that pseudo orbits g1 , g2 ∈ P (δ) are η-rep-close if there exists a reparametrization γ ∈ Rep such that d(g1 (t), g2 (γ(t))) < η. We will use the following properties of the map T . By shadowing and expansivity we know that for sufficiently small δ that for any g ∈ P (δ) and t ∈ R there exists t0 ∈ R such that T ◦ σt (g) = φt0 ◦ T (g). Fix small enough η, δ > 0 such that if g1 , g2 ∈ P (δ) are η-rep-close then there exists |τ | < τ0 such that (17)
T g2 = φ(τ, T g1 ).
Fix δ1 > 0 sufficiently small such that V0 = B(δ1 , Λ) ⊂ U . For any point z ∈ Iφ (V0 ) and t ∈ R let us choose a point g(t) ∈ Λ such that the inclusion φt (z) ∈ L(δ1 , g(t)) holds. We also assume that δ1 is sufficiently small so that the map g(t) is a δ-pseudo orbit. Define a map S : V0 → P (δ) as S(z) := g. We would like to emphasize that the choice of g(t) is not unique, however for any such choice T (g) = z. Again from the expansivity property we have T ◦ S = id. Also, for ε > 0 sufficiently small there exists Θ > 0 such that if g1 , g2 ∈ P (δ) and d(g1 (t), g2 (t)) < δ, |t| < Θ then there exists |τ | < τ0 d(S(g1 ), φ(τ, S(g2 ))) < ε. For δ possibly smaller we can fix η > 0 sufficiently small so that for any g ∈ P (δ) pseudo orbits g and S ◦ T g are η-rep-close. Additionally,
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there exists a neighborhood V1 ⊂ V0 of Λ such that for any x ∈ Iφ (V1 ) the inclusion S(x) ∈ P (δ) holds. Similarly, fix δ00 , δ10 , δ 0 , η 0 > 0, maps T 0 : P (δ00 ) → M 0 and S 0 : B(δ10 , Λ0 ) → P (δ 0 ), and a neighborhood V10 of Λ0 . Now we are ready to construct the maps h1 , h01 . Let us choose δ2 ∈ (0, δ) such that for any g ∈ P (δ2 ) the inclusion h(g) ∈ P (δ 0 ) holds. Now let us choose V ⊂ V1 an open neighborhood of Λ such that for any x ∈ Iφ (V ) the inclusion S(x) ∈ P (δ2 ) holds. Notice that if g is a pseudo-orbit contained strictly in Λ, then we can define a pseudoorbit in Λ0 by h(g). We now define the map h1 : Iφ (V ) → M 0 as h1 := T 0 ◦ h ◦ S. Similarly, define V 0 and map h01 : IX 0 (V 0 ) → M as h01 = T ◦ h−1 ◦ S 0 . Among the properties of maps h1 , h01 the most difficult is relation (15). We will give its proof in full details. Properties (13) and (14) will follow directly from the definitions of the functions h1 and h01 . Relations (8)–(12) can be easily deduced (decreasing V and V 0 if necessarily) from • properties described above, • expansivity of the vector fields in V, V 0 , and • continuity of h, h−1 . We prove only the second equality in (15). Note that h01 ◦ h1 = T ◦ h−1 ◦ S 0 ◦ T 0 ◦ h ◦ S. For η 0 > 0 perhaps smaller we know that if g10 and g20 are η 0 -rep-close then h−1 g10 , h−1 g20 are η-rep-close. Also, for δ 0 > 0 perhaps smaller we know that if g10 ∈ P (δ 0 ), then pseudo orbits g10 and S 0 ◦ T 0 g10 are η 0 -rep-close. By the continuity of h for V , perhaps smaller, we know that for any x ∈ Iφ (V ) the inclusion h ◦ Sx ∈ P (δ 0 ) holds. Let x ∈ V . Set g1 := Sx, g10 := hg1 , x0 := T 0 g10 , g20 := S 0 x0 , g2 := h−1 g20 , y := T g2 . By construction of V we know g10 ∈ P (δ 0 ). Note that g20 = S 0 ◦ T 0 g10 . Hence, g10 and g20 are η 0 -rep-close. Also, we know that g1 = h−1 g10 , and g2 = h−1 g20 . Hence, g1 and g2 are η-rep-close. Finally, we have x = T g1 ,
y = T g2 .
Now we know that (17) implies the second equality (15).
Proof of Theorem 1.6. We prove that if Λ is premaximal then Λ0 is premaximal. The converse statement can be proven similarly. Consider neighborhoods V ⊂ U of Λ and V 0 ⊂ U 0 of Λ0 and maps h1 , h01 from Theorem 4.4.
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Assume that Σ ⊂ V is a locally maximal set with isolating neighborhood W = B(η, Σ), such that Σ0 = O(h1 (Σ)) ⊂ V 0 . Below we will prove that Σ0 is locally maximal. Consider η1 > 0 such that for x ∈ B(η1 , Σ) and |τ | < τ0 the inclusion φτ (x) holds. Using equality (14) and inequality (10) for ε = η1 we find η10 > 0 such that if x0 ∈ W 0 := B(η10 , Σ0 ) then h01 (x0 ) ∈ B(η1 , Σ). Let us prove that Σ0 = Iφ0 W 0 . Assume contrarily that there exists 0 x ∈ W 0 \ Σ0 such that O(x0 ) ⊂ W 0 . Relations (13), (14), (15) imply that [ (18) O(h(x0 )) = φτ (h01 (x01 )). |τ | 0 any δ-pseudo orbit consisting of points from Λ can be shadowed by a trajectory from Λ for small enough δ). It easily leads us to the following. Proposition 4.6. If sh(Λ, δ) = Λ for some δ > 0. Then Λ is locally maximal. As a reminder if Λ is a locally maximal set, then for ε > 0 sufficiently small we know that Bε (Λ) is an isolating set for Λ. In [9] Crovisier provides an example of a hyperbolic set that is never included in a locally maximal one, and this example shows that to establish premaximality it is not enough to say that a hyperbolic set is included in a locally maximal one or that the shadowing closure stabilizes for some δ > 0.
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T. FISHER, T. PETTY, AND S. TIKHOMIROV
To be more precise, let Let A and B be 2 × 2 hyperbolic toral automorphisms such that A has two fixed points p and q and the hyperbolicity in A dominates the hyperbolicity in B. Let F be a diffeomorphism on the 4-torus defined by F0 (x, y) = (Ax, By) and fix r a fixed point of B. Crovisier proves that if we let V be a sufficiently small neighborhood of (q, r), then the set \ Λ= f n (T4 − V ) n∈Z
is a hyperbolic set and the only locally maximal hyperbolic set that Λ can be included in is the entire manifold T4 ; so this example is not premaximal. However, Λ is included in a locally maximal hyperbolic set and for δ > 0 sufficiently large may be in a shadowing closure that stabilizes. Proof of Theorem 1.7. We first consider the case of diffeomorphisms. Suppose that Λ is a hyperbolic set for a diffeomorphism f : M → M and suppose that for any neighborhood U of Λ there exists a δ > 0 such that the shadowing closure of Λ stabilizes inside U . Now the previous proposition shows that the stabilizer is a locally maximal set and Λ is premaximal. To prove the other implication let f : M → M be a diffeomorphism and let Λ be a premaximal set for f and V be a neighborhood of Λ. Then there exists some η > 0 such that [ Bη (Λ) = Bη (x) ⊂ V x∈Λ
and Λη = If (Bη (Λ)) is hyperbolic. ˜ be a locally maximal hyperbolic set such that Let Λ ˜ ⊂ Λη ⊂ Bη (Λ) ⊂ V. Λ⊂Λ ˜ is an isolating neighborhood of Λ ˜ and Fix ε ∈ (0, η) such that Vε (Λ) ˜ is ε shadowed in Λ ˜ fix δ ∈ (0, ε/2) such that every δ-pseudo orbit in Λ ˜ by a unique point in Λ. From the choice of constants above we know that Λ ⊂ sh(Λ, δ) ⊂ Λε . Furthermore, we know that each δ-pseudo orbit of Λ is a δ-pseudo orbit ˜ so there exists a unique point y ∈ Λ ˜ that is a ε-shadowing point of Λ ˜ Let Λ1 = sh(Λ, δ). of the pseudo orbit. Hence, sh(Λ, δ) ⊂ Λ. If Λ1 = Λ we know from the previous proposition that Λ is locally maximal. So suppose that Λ1 6= Λ and let ν1 = dH (Λ1 , Λ) where dH (·, ·) is the Hausdorff distance between the sets. More generally, let Λj+1 = sh(Λj , δ), νj+1 = dH (Λj+1 , Λj ) for j ∈ N. By the shadowing estimates we know that νj ∈ (0, ε).
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Claim 4.7. There exists γ > 0 such that for all j ∈ N if Λj 6= Λj+1 and Λj+1 6= Λj+2 , then either νj ≥ γ or νj+1 ≥ γ. Proof of Claim. Consider γ ∈ (0, δ/4) such that for all x, y ∈ M if d(x, y) < γ, then d(f (x), f (y)) < δ/4 and d(f −1 (x), f −1 (y)) < δ/4. Suppose that Λj 6= Λj+1 and Λj+1 6= Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . We will construct a δ-pseudo orbit in Λj that y ε-shadows. This will show that y ∈ Λj+1 , a contradiction. Then there exists some y0 ∈ Λj+1 and x0 ∈ Λj such that d(y, y0 ) < νj+1 and d(y0 , x0 ) < νj < γ. Then d(f (y), f (y0 )) < δ/4 and d(f (y0 ), f (x0 )) < δ/4. Also, we know there exists points y1 ∈ Λj+1 and x1 ∈ Λj such that d(f (y), y1 ) < νj+2 and d(y1 , x1 ) < νj+1 . Hence, d(f (x0 ), x1 ) ≤ d(f (x0 ), f (y0 )) + d(f (y0 ), f (y)) + d(f (y), y1 ) + d(y1 , x1 ) ≤ δ/4 + δ/4 + νj+2 + νj+1 < δ and d(f (y), x1 ) ≤ d(f (y), y1 ) + d(y1 , x1 ) < νj+2 + νj+1 < δ/2 < ε. Continue inductively, we can construct a forward δ-pseudo orbit (xk )∞ k=0 such that y ε-shadows the pseudo orbit. Also, since the estimates on γ apply for f −1 we can construct a bi-infinite δ-pseudo orbit (xk ) in Λj such that y ε shadows (xk ). Then y ∈ Λj+1 , a contradiction. We now return to the proof of the theorem. ˜ for all Repeating the above arguments for Λj we see that Λj ⊂ Λ j ∈ N. We know from Proposition 4.6 that if Λj+1 = Λj for some j ∈ N that Λj is locally maximal. To conclude the proof of the theorem we simply need to show that the sequence Λj stabilizes. Suppose that the sequence Λj does not stabilizer. We know from the above claim that for each j the sets Λj+1 or Λj+2 will be a distance of γ from the set Λj using the Hausdorff metric. Inside a compact metric space we know an increasing sequence of compact sets converges in the Hausdorff topology. Hence, if the sequence does not stabilize there exists some N ∈ N where the Hausdorff distance from ΛN to Λ is greater ˜ and Λ ˜ ⊂ Vη (Λ). than η. This is a contradiction since each Λj ⊂ Λ Theorem 1.7 is proved for the case of diffeomorphisms. The proof for the case of flows follows the same ideas. Below we provide a detailed proof of the Claim 4.7, which is the central step of the proof of Theorem 1.7.
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T. FISHER, T. PETTY, AND S. TIKHOMIROV
We will use the following construction. For a sequence of points {xk }k∈Z consider a map g{xk } : R → M defined as g{xk } (t) := φ(t − [t], x[t] ), where [t] is the integer part of t. Consider map r : M → M defined as r(x) = φ(1, x). Note that r does not necessarily satisfy the shadowing property. We will use the following. F1: There exists ε1 > 0 such that for any sequence {xk } and point y satisfying (19)
d(xk , f k (y)) < ε1 ,
k∈Z
the following inequality holds d(g{xk } (t), φ(t, y)) < ε. Without loss of generality we can assume that δ < ε1 . F2: There exists δ1 > 0 such that if {xk } with xk ∈ Λj is a δ1 -pseudo orbit of the map r then g{xk } ⊂ Λj is a δ-pseudo orbit of the flow φ. Consider γ ∈ (0, δ1 /4) such that for all x, y ∈ M if d(x, y) < γ, then d(r(x), r(y)) < δ1 /4 and d(r−1 (x), r−1 (y)) < δ1 /4. As in the case of diffeomorphisms suppose that Λj 6= Λj+1 and Λj+1 6= Λj+2 and νj , νj+1 ∈ (0, γ). Fix y ∈ Λj+2 − Λj+1 . Arguing similarly to the case of diffeomorphisms we can construct a δ1 -pseudo orbit {xk } ⊂ Λj of the map r which satisfies (19). Properties F1 and F2 implies that δ-pseudo orbit g{xk } is ε-shadowed by y. This shows that y ∈ Λj+1 , a contradiction. References [1] V. M. Alekseev. Symbolic dynamics. Eleventh Mathematical School (Summer School, Kolomyya, 1973), pp. 5-210. Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1976. [2] D. V. Anosov. Intrinsic character of one property of hyperbolic sets. J. Dyn. Control Syst. 16, no. 4, 485-493, 2010. [3] D. V. Anosov. Extension of zero-dimensional hyperbolic sets to locally maximal ones. Math. Sb. 201, no. 7, 3-14, 2010. [4] D. V. Anosov. On certain hyperbolic sets. Mat. Zametki, 87, no. 5, 650-668, 2010. [5] D. V. Anosov. Local maximality of hyperbolic sets. Proc. Steklov Inst. Math. 273, no. 1, 23-24, 2011. [6] D. V. Anosov. On trajectories located entirely near a hyperbolic set. Sovrem. Mat. Fundam. Napravl. 45, 5–17, 2012.
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[7] V. Ara´ ujo, V. and M. J. Pac´ıfico, Three dimensional flows, Publica¸c˜oes Matem´ aticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matem´ atica Pura e Aplicada (IMPA), Rio de Janeiro, 2007. [8] K. Burns and K. Gelfert, Lyapunov spectrum for geodesic flows of rank 1 surfaces, Discrete Contin. Dyn. Syst., 34 (2014), no. 5, 1841-1872. [9] S. Crovisier. Une remarque sur les ensembles hyperboliques localement maximaux. C. R. Math. Acad. Sci. Paris, 334, no. 5, 401-404, 2001. [10] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J., 21 (1971/1972), 193-226. [11] N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math. J., 23 (1973/74), 1109-1137. [12] N. Fenichel, Asymptotic stability with rate conditions. II, Indiana Univ. Math. J., 26 (1977), no. 1, 81-93. [13] T. Fisher. Hyperbolic sets that are not locally maximal. Ergod. Th. Dynamic. Systems, 26, no. 5, 1491-1509, 2006. [14] T. Hunt, Low Dimensional Dynamics: Bifurcations of Cantori and Realisations of Uniform Hyperbolicity, PhD Thesis, University of Cambridge, Queen’s College, 2000. [15] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995). [16] Y. Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, 16 Birkh¨ auser Boston, Inc., Boston, MA, 1988. [17] T. Petty, Nonlocally maximal hyperbolic sets for flows, Maters Thesis, Brigham Young University, 2015. [18] S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Math., 1706, Springer, Berlin, 1999. T. Fisher, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected] T. Petty, Department of Mathematics, Brigham Young University, Provo, UT 84602 E-mail address: [email protected] S. Tikhomirov, Max Planck Institute for Mathematics in the sciences, Inselstrasse 22, 04103, Leipzig, Germany; Chebyshev Laboratory, Saint-Petersburg State University, 14th Line 29B, Vasilyevsky Island, St.Petersburg 199178, Russia E-mail address: [email protected]
