LIVSIC THEORY FOR COMPACT GROUP EXTENSIONS OF ...
MOSCOW MATHEMATICAL JOURNAL Volume 5, Number 1, January–March 2005, Pages 55–66
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC OF HYPERBOLIC SYSTEMS DMITRY DOLGOPYAT Dedicated to Yu. S. Ilyashenko on the occasion of his 60th birthday
Abstract. We prove Livsiˆc-type theorems for rapidly mixing compact group extensions of Anosov diffeomorphisms. 2000 Math. Subj. Class. 3730C, 37D30, 37J40. Key words and phrases. Cocycle equation, transfer operator, partial hyperbolicity, small divisors.
1. Introduction In recent years, several new phenomena in dynamics were discovered by looking at small perturbations of compact group extensions of hyperbolic systems [8], [14]. In view of this, it is desirable to develop a general theory of perturbations of such systems. The first step towards this goal is to understand infinitesimal perturbations, that is, study homological equations over such systems. In this paper, we study the regularity of solutions to cocycle equations. Regularity theory plays an important role in rigidity theory. Two of the most studied cases are translations of Td and Anosov diffeomorphisms (see [7], [11], [15] for the analysis of some other systems). The systems considered in this paper exhibit a mixture of hyperbolic and elliptic behaviors. Let M be a compact C ∞ Riemannian manifold, and let f : M → M be an Anosov diffeomorphism. Suppose that G is a compact connected Lie group, H a Lie subgroup of G, Y = G/H, and τ ∈ C ∞ (M, G). Let N = M × Y . We define F : N → N by F (x, y) = (f x, τ (x)y). A function A on N is called a coboundary if A = B − B ◦ F.
(1)
If B is bounded, H¨ older, smooth, etc., we say that the coboundary A is bounded, H¨older, smooth, etc. Take φ ∈ C α (M ); let µφ be the Gibbs state with potential φ, and let dνφ = dµφ dy. In this paper, we prove the following theorem. Take z0 ∈ N . Received May 14, 2003. c
2005 Independent University of Moscow
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D. DOLGOPYAT
Theorem 1. Let F be rapidly mixing. If A ∈ C ∞ (N ) is a coboundary in L2 (νφ ) for some H¨ older function φ, then A is a C ∞ coboundary. In particular, if A is a bounded coboundary, then it is C ∞ . Moreover, there exists a k0 such that, if A belongs to C k (N ), then B belongs to C k−k0 (N ). If B satisfies the normalization condition B(z0 ) = 0, then kBkk−k0 ≤ ConstkAkk . (2) We refer the reader to the next section for the definition of rapid mixing. Recall [4] that a generic extension is rapidly mixing. Observe that Theorem 1 implies, in particular, that the set of coboundaries is closed in C k for k > k0 . We also present versions of this theorem for extensions of subshifts ofR finite type. Our results are also true for relative coboundaries. Let A0 (x) = A(x, y)dy. We say that A is a relative coboundary if A = A0 + B − B ◦ F.
(3)
Theorem 2. The assertions of Theorem 1 are valid for relative coboundaries. 2. Preliminaries 2.1. Subshifts of finite type. Here we present some results about subshifts of finite type and their compact extensions. The proofs can be found in [13, Chapters 3 and 8]. For a geometric interpretation of the results about extensions, see, e. g., [4, Section 2]. Let a be a finite alphabet; by A we denote a Card(a) × Card(a)-matrix whose entries are zeroes and ones. Suppose that Σ = ΣA is the associated (twosided) subshift of finite type, that is, Σ = {{ωi }+∞ i=−∞ : ωi ∈ a and Aωi ,ωi+1 = 1}. Let σ be the shift defined by σ(ω)i = ωi+1 . Given θ < 1, we consider the metric dθ on Σ given by dθ (ω 0 , ω 00 ) = θj , where j = max(k : ωi0 = ωi00 for |i| < k). Let Cθ (Σ) denote the set of dθ -Lipschitz functions. Given φ ∈ Cθ (Σ), we denote the Gibbs measure with potential φ by µφ , that is, hµφ + µφ (φ) = sup(hµ + µ(φ)), µ
where the supremum is over all σ-invariant probability measures. We shall use the fact that homologous functions have the same Gibbs measures. Let G be a ˜ = Σ × Y and compact connected Lie group, and let τ ∈ Cθ (Σ, G). We set N ˜ ˜ ˜ define F : N → N by F (ω, y) = (σω, τ (x)y). Consider the measure νφ given by ˜ ) = Cθ (Σ, C k (G)). We say that F˜ is rapidly mixing if, dνφ = dµφ dg. Let Ck,θ (N for any φ and N , there exists a k such that νφ (A(ω, y)B(F˜ n (ω, y))) − νφ (A)νφ (B) ≤ ConstkAkk,θ kBkk,θ n−N for all A, B ∈ Ck,θ . It was shown in [4, Theorem 4.3] that rapid mixing is generic among compact group extensions of subshifts of finite type. Let Σ+ be the associated one-sided subshift, which is defined similarly to Σ + but for the one-sided sequence ω = {ωi }∞ i=0 . The set Cθ (Σ ), Gibbs states, rapid
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC
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mixing, etc. are defined for one-sided shifts as for two-sided shifts. Suppose that ˜ →N ˜ is the skew extension defined by τ ∈ Cθ (Σ, G), φ ∈ Cθ (Σ) is a potential, F˜ : N and A ∈ Ck,θ is an observable. Then there are τ ∗ ∈ C√θ (Σ, G), M ∈ C√θ (Σ, G), φ∗ ∈ Cθ (Σ+ ), ψ ∈ C√θ (Σ), A∗ ∈ Ck,√θ (Σ+ ), and K ∈ Ck,√θ (Σ) such that τ ∗ = (M ◦ σ)τ M −1 ,
φ∗ = φ + ψ − (ψ ◦ σ),
A∗ = A + K − K ◦ F˜ .
Moreover, φ∗ can be chosen so that X ∗ eφ ($) = 1 for any ω.
(4)
σ$=ω
The skew products defined by τ and τ ∗ are conjugate, φ and φ∗ have the same Gibbs measure, and A is a coboundary if and only if A∗ is a coboundary. ˜ + = Σ+ × Y . Suppose that F˜ is the skew extension determined by some Let N τ ∈ Cθ (Σ+ , G), ∆ is a G-invariant Laplacian on G, and Hλ = {ϕ : ∆ϕ = λϕ}. We endow Hλ with the L2 -norm. We set Cλ,θ (Σ+ ) = Cθ (Σ+ , Hλ ). Let φ be any H¨older function on Σ+ such that X eφ($) = 1 for any ω, (5) σ$=ω
and let µφ be the Gibbs measure for φ. Consider the transfer operator X (L(h))(ω, g) = eφ($) h($, τ −1 ($)g).
(6)
σ$=ω
It preserves Cλ,θ (Σ+ ). Let Lλ denote the restriction of L to Cλ,θ (Σ+ ). Proposition 1 [4, Proposition 4.4]. If F˜ is rapidly mixing, then there exist a C and an s such that n 1 n s kLλ k ≤ Cλ 1 − . (7) Cλs 2.2. Anosov diffeomorphisms. Recall that a diffeomorphism F : M → M is said to be Anosov if there exist an f -invariant splitting T M = Es ⊕ Eu and constants C, ρ < 1 such that kdf n vk ≤ Cρn kvk for any v ∈ E s
and kdf −n vk ≤ Cρn kvk for any v ∈ E u .
The distributions E s and E u are uniquely integrable, they are tangent to the foliations W s and W u , respectively. Since W s and W u are transverse, if x, y ∈ M u s are close to each other, then the intersection Wloc (x) ∩ Wloc (y) consists of one point, which we denote by [x, y]. A set Π is called a parallelogram if [x, y] ∈ Π for all x, y ∈ Π. A partition Π = {Π1 , Π2 , . . . Πn } is said to be Markov if, for all x ∈ Int(Πi ), f WΠs (x) ∈ WΠs (f x), f −1 WΠu (x) ∈ WΠu (f −1 x),
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D. DOLGOPYAT
∗ where WΠ∗ (z) = Wloc (z) ∩ Πj for z ∈ Πj . Given a Markov partition Π, we can consider the subshift of finite type Σ for which a = {1, 2 . . . n} and Aij = 1 if and only if f (IntPi ) ∩ Pj = 0. The map ζ : Σ → M given by \ ζ(ω) = f −j Πωj j
defines a semicongugacy between σ and f . For a function τ from M to G, let τ¯ = τ ζ. Then ζ × id is a semicongugacy between F (x, y) = (f x, τ (x)y) and F˜ (ω, y) = (σω, τ¯(ω)y). We shall use the fact that the skew extension F is partially hyperbolic. That is, there exist an F -invariant splitting T N = EFs ⊕ EFc ⊕ EFu and constants C, ρ < 1 such that kdF n vk ≤ Cρn kvk for any v ∈ EFs ,
kdF −n vk ≤ Cρn kvk for any v ∈ EFu
and EFc is the tangent space to the fibers. The definition of Gibbs states for f is similar to that for σ. An important special case is the so-called SRB measure, which is the Gibbs measure with potential φSRB = − ln det(df |E u ). The importance of the SRB measure comes from the fact that, if Φ ∈ C(M ), then n−1 1X Φ(f j x) → µSRB (Φ) n j=0
as n → +∞
for Lebesgue almost all x. For any α, there exists θ such that if φ ∈ C α (M ), then φ¯ = φ ◦ ζ ∈ Cθ (Σ); µφ is a Gibbs state for f if and only if µφ¯(Ω) = µφ (ζ(Ω))
for any Ω ⊂ Σ.
We say that F is rapidly mixing if for any φ and N , there exists k such that νφ (A(x, y)B(F n (x, y))) − νφ (A)νφ (B) ≤ ConstkAkk kBkk n−N for any A, B ∈ C k (M ). The extension F is rapidly mixing if and only if F˜ is rapidly mixing. 3. Symbolic Systems ˜+ → N ˜ + is a rapidly mixing 3.1. One-sided shifts. In this subsection, F˜ : N + + extension of the one-sided subshift. Let Cr,θ (Σ ) = Cθ (Σ , C r (Y )). Lemma 1. Let A ∈ C∞,θ (Σ+ ) be an L2 (νφ )-coboundary for some φ ∈ Cθ (Σ+ ) such that A = B − B ◦ F˜ , where B ∈ L2 (νφ ). Then B has a version in C∞,θ (Σ+ ). Moreover, there exists a k0 such that, if A ∈ Ck,θ (Σ+ ), then B ∈ Ck−k0 ,θ (Σ+ ) and kBkk−k0 ,θ ≤ Const(k)kAkk,θ .
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Proof. According to what R we can assume that φ satisfies P was said in Section 2.1, (5). Let A = A0 + λ6=0 Aλ , where A0 (ω) = A(ω, g)dg and Aλ ∈ Hλ . Let P B = λ Bλ . Since F commutes with projections to Hλ , we have Aλ = Bλ − Bλ ◦ F˜ .
(8) +
In particular, A0 = B0 − B0 ◦ σ and, according to [13], B0 ∈ Cθ (Σ ). Hence we can assume without loss of generality that A0 ≡ 0. Applying Lλ to (8), we obtain Lλ Aλ = (Lλ − 1) Bλ . Thus, Bλ = −(1 − Lλ )−1 Lλ Bλ . There exists a p = p(G) such that Const kAλ kλ ≤ k/2−p kAkk,θ . λ By Proposition 1, there exists an s such that k(1 − Lλ )−1 k ≤ Const λs . Hence kBλ kλ ≤ Const λ2s kAλ kλ ≤
Const λk/2−(2s+p)
We have ¯ kBλ kk−k0 ,θ ≤ Const λp+
Let B =
k−k0 2
(9)
(10) kAkk,θ .
kBλ kλ .
(11) (12)
P
Bλ . Then X X ¯ 0 /2) kBkk−k0 ,θ ≤ kBλ kk−k0 ,θ ≤ Const λp+p+2s−(k kAkk,θ , λ
λ
λ
and this series converges if k0 is large enough. This completes the proof.
˜ →N ˜ be an extension of the two-sided subshift 3.2. Two-sided shifts. Let F˜ : N of finite type. Lemma 2. Let A = B − B ◦ F˜ . If A ∈ C∞,θ (Σ), then B ∈ C∞,θ1/4 (Σ). Moreover, there exists a k0 such that, if A ∈ Ck,θ (Σ), then B ∈ Ck−k0 ,θ1/4 (Σ) and kBkk−k0 ,θ1/4 ≤ Const(k)kAkk,θ . Proof. Let τ ∗ = (M ◦ σ)τ M −1 . Then the change of variables y ∗ = M y conjugates F˜ and F ∗ (ω, y ∗ ) = (σω, τ ∗ (ω)y ∗ ). Thus, A is an F˜ -coboundary if and only if A∗ (ω, y ∗ ) = A(ω, M −1 y ∗ ) is an F ∗ -coboundary. Let us represent A∗ as A∗ = A∗∗ +K ∗ −K ∗ ◦F ∗ , where A∗∗ ∈ Ck,θ1/4 (Σ+ ). Then A∗ is an F ∗ -coboundary if and only if A∗∗ is. But by Lemma 1, A∗∗ = B ∗∗ − B ∗∗ ◦ F , where B ∗∗ ∈ Ck−k0 ,θ1/4 (Σ+ ). Thus, A = (B + K) − (B + k) ◦ F˜ , where B(ω, y) = B ∗∗ (ω, y) and K(ω, y) = K ∗ (ω, M y). This proves the lemma. Corollary 1. If ω is a periodic orbit of σ, say σ n ω = ω, then n−1 X j Aλ (F˜ (ω, y)) ≤ Cλs kAλ kθ,k0 d(τn (ω)y, y). j=0
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D. DOLGOPYAT
Proof. We have
n−1 X √ j Aλ (F˜ (ω, y)) ≤ |Bλ (ω, τn (ω)y) − Bλ (ω, y)| ≤ λ d(τn (ω)y, y)kBλ k, j=0
and the required assertion follows from (11).
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4. Anosov Diffeomorphisms 4.1. H¨ older continuity. We proceed to prove Theorem 1. In this section, we shall show that B has a H¨ older version. Take a Markov partition Π of M . Let Σ be the associated subshift of finite type, and let ζ : Σ → M be the semiconjugacy ζ ◦ σ = f ◦ ζ. ¯ −B ¯ ◦ F˜ , and We set A¯ = A ◦ ζ, A¯ = B Consider τ¯ and F˜ defined as in Section 2.2. P −1 ¯=P B ¯λ . Let Bλ = B ¯λ ◦ ζ −1 and B = B B . Since ζ is discontinuous, we λ λ λ cannot assert that the Bλ are H¨older, but we can state the following lemma, which is a consequence of the periodic leaf estimates of Corollary 1. Suppose p = (x, y) has a dense orbit. We have n−1 X Bλ (F n p) = Bλ (p) − Aλ (F j p). j=0
Lemma 3. Bλ |Orb(x,y) is uniformly H¨ older continuous with H¨ older constant CkAλ kλs . Proof. Suppose that m < n and d(F n p, F m p) ≤ ε. We set k = n − m, z = f m x, and q = F m p. By the Anosov closing lemma, there exists an x ˜ ∈ M such that f k x ˜=x ˜ and d(f j z, f j y) ≤ Cd(f k z, z)γ ρmax(j,k−j) for some γ > 0 and ρ < 1. Let u = (˜ x, y). Then k−1 k−1 k−1 X X X j j j j k Bλ (f q) − Bλ (q) = Aλ (F u) + [Aλ (F q) − Aλ (F u)] . Aλ (F q) ≤ j=0
j=0
j=0
By Corollary 1, the first part is O(kAλ kλs4 dα (p, F k p)) and the second part is
√ O(kAλ k λdγ (p, F k p)), √ because Aλ is Lipschitz with constant λkAλ k.
Since Orb(p) is dense, we can extend the Bλ to H¨older functions on N . Lemma 4. Under the conditions of Theorem 1, the restriction of B to each fiber is smooth. Moreover, there exists a k0 such that kBkC α0 (M,C k−k0 (G)) ≤ Const(k)kAkC α (M,C k (G)) . Proof. First, let us show that B has a H¨older version. By Lemma 3, each Bλ admits an extension from Orb(x, g) to N which is H¨older with H¨older norm at most ConstkAλ kλs . By the continuity of Aλ , this extension satisfies Aλ = Bλ − Bλ ◦ F , and Const kAλ k ≤ (k/2)−p kAkC k (N ) . λ
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D. DOLGOPYAT
For B =
P
λ
Bλ , we have !
kBkC α (N ) ≤
X
kBλ kC α (N ) ≤ Const
λ
X
λ
p+s−k/2
kAkC k (N ) ,
λ
and this series converges if k is large enough. In other words, there exists a k1 such that BC α (N ) ≤ ConstkAkC k1 N . Applying this to ∆m A, we obtain kBkC α (M,H2m (G)) ≤ Constk∆m BkC α (N ) ≤ Constk∆m AkC k1 (N ) ≤ ConstkAkC k1 +2m (N ) and the application of the Sobolev embedding theorem completes the proof.
4.2. Smoothness. In this section, we prove the smoothness of B in the transverse directions. Lemma 5. Restrictions of B to the leaves of WFs and WFu are smooth. Proof. It suffices to consider WFs . We have A(p) = B(p) − B(F p). Thus, B(p) = A(p) + B(F p). Hence if p ∈ W s (p0 ) then B(p) − B(p0 ) =
∞ X
[A(F p j) − A(F j p0 )].
j=0
Since F j are contractions on WFS , this series can be differentiated term by term arbitrarily many times (see [3]). Now, we need the following fact [10]. Proposition 2 (Journe lemma). Let F1 and F2 be two continuous transverse foliations with smooth leaves. If B is a continuous function whose restrictions to the leaves of F1 and F2 are smooth, then B is smooth. Moreover, there exists a k0 such that, if the restrictions of B to the leaves are C k , then B is C k−k0 . This proposition, together with Lemmas 4 and 5, implies that B is smooth on each leave of W sc ; since it is also smooth on each leave of W u , we conclude that B is smooth. This completes the proof of Theorem 1. Remark. The weaker versions of the Journe lemma proven in [3], [9] are sufficient for the proof. 4.3. Relative coboundaries. Proof of Theorem 2. Apply Theorem 1 to ∆A. 4.4. A counter-example. Take G = T2 and F (x, t1 , t2 ) = (f x, t1 + α1 r(x), t2 + α2 r(x)). Suppose that α1 /α2 is irrational and, for all N , there exist m1,N , m2,N ∈ Z such that |α1 m1,N + α2 m2,N | ≤ m−N 2,n .
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC
Changing the indexation if necessary, we can assume that m2,N > N 2 . ΦN (x, t1 , t2 ) = exp(2πi(m1,N t1 + m2,N t2 )). Then
63
Let
Φ ◦ F = exp(2πi(m1,N α1 + m2,N α2 )r(x))ΦN . Let AP= N ((ΦN − Φ ◦ F )/N 2 ). Then A ∈ C ∞ (N ) and A = B − B ◦ F m where B = N ΦN ∈ C 0 (N ) − C 1 (N ). Considering suitable linear combinations of ΦN , we see that F is not rapidly mixing. This shows that Theorem 1 is not valid for arbitrary extensions. P
4.5. Obstructions. In this section, we give criteria for a function to be a coboundary. Most of these criteria come from other papers, however their applicability is a consequence of the fact that different notions of coboundaries coincide in the situation under consideration. Sometimes, it is easier to verify that A is a relative coboundary. It is sufficient, because it is well known when A0 is an f -coboundary. (i) We set Dφ (A) = νφ (A2 ) − νφ2 (A) + 2
∞ X
[νφ (A(A ◦ F˜ j ))νφ2 (A)].
j=1
Proposition 3 [6]. A is cohomologous to a constant if and only if there exists a φ such that Dφ (A) = 0, or, equivalently, Dφ (A) = 0 for any ψ. Proof. Without loss of generality, we can assume that νφ (A) = 0. Then Dφ (A) = 0 if an only if A is an L2 (νφ )-coboundary (by the spectral theorem). The latter condition is equivalent to A being a H¨older coboundary (by Theorem 1), which is, in turn, equivalent to A being an L2 (νψ )-coboundary for each ψ. (ii) Let P = {p0 , p1 . . . pn } be a chain such that pk+1 ∈ W s (pk ) ∪ W u (pk ). We say that P is closed if p0 = pn . We set X r(P ) = r(pk , pk+1 ), k
where
r(pk , pk+1 ) =
∞ X A(F j pk+1 ) − A(F j pk )
if pk+1 ∈ W s pk ,
j=0
−1 X A(F j pk ) − A(F j pk+1 ) if pk+1 ∈ W s pk . j=−∞
The following statement is Corollary 3.1 from [11]. Proposition 4. If F has the accessibility property, then A is cohomologous to a constant if and only if r(P ) = 0 for any closed chain P . (iii) The next result follows from the proof of Theorem 1. Proposition 5. A is a relative coboundary if and only if ∆N A is a coboundary for any N ∈ N.
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D. DOLGOPYAT
(iv) Let G = T, and let ν be the SRB measure for F . Consider the one-parameter family Fε (x, z) = (f x, z + τ (x) + εA(x, z) + ε2 α(ε, x, z). Suppose that νε is a u-Gibbs measure for Fε , that is, the projection of νε to M is µSRB . Proposition 6. A is a relative coboundary if and only if λc (νε ) = o(ε2 ). Proof. We use the asymptotics [5] νε (H) = ν(H) + εω(H) + o(ε), where ω(H) =
∞ X dH . ν A ◦ F −j dz j=1
ε We want to apply this to Hε = ln dF dz . We have
So
dFε dA dα(0, x, z) =1+ε + ε2 + o(ε2 ). dz dz dz " 2 # dA 1 dA dFε 2 dα(0, x, z) =ε +ε − ln + o(ε2 ). dz dz dz 2 dz
Hence dA dFε νε ln = εν dz dz " 2 X # ∞ 2 1 dA dα(0, x, z) −j d A 2 ν A◦F − ν + + o(ε2 ). +ε ν 2 dz 2 dz dz j=1 Since dν = dµSRB dg, it follows that dα(0, x, z) dA =ν = 0 and ν dz dz d2 A dA dA ν A ◦ f −j 2 = −ν ◦ F −j . dz dz dz Hence " 2 X # ∞ dA dA dFε 2 1 −j dA ∼ −ε ν − ν ◦F , νε ln dz 2 dz dz dz j=1 that is, ε2 DSRB ( dA dFε dz ) νε ln ∼− . dz 2
(13)
Therefore, λc (νε ) = o(ε2 ) dA if and only if DSRB ( dA dz ) = 0. By Proposition 3, dz is a coboundary, which means that A satisfies (3).
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Appendix A. The Non-Mixing Case Note that Theorem 1 may be valid for an Anosov time rotation even though it is not mixing. In this appendix, we give an extension of Theorem 1 to the nonmixing case. In order to explain the result, we recall some background. Given ¯ = M × G and consider the principal extension F¯ : N ¯ → N ¯ given by τ , let N ¯ F (x, g) = (f (x), τ (x)g). Recall the definition of Brin groups [1], [2]. Given a partially hyperbolic diffeomorphism, we call a sequence P = {p1 , p2 , . . . , pn } a e-chain (respectively t-chain) if pj+1 ∈ W u (pj ) ∪ W s (pj ) (respectively, pj+1 ∈ W u (pj ) ∪ W s (pj ) ∪ Orb(pj )). Take a reference point x ∈ M . Given any chain ¯ starting at P ⊂ M with xn = x1 = x and any g1 ∈ G, there is unique chain P¯ ⊂ N ¯ (x, g1 ) and covering P . P is not closed, instead we have gn = g(P )g1 . Let Γt (x) (Γe (x)) denote the set of all g(P ) for all closed t-chains (respectively, e-chains) starting at x. Proposition 7 (Brin, [1], [2]). (a) The Γ∗ (x) are groups. The Γ∗ of different points are conjugate, Γt is a normal subgroup of Γe , and Γe /Γt is cyclic. In particular, ¯ e /Γ ¯ t is Abelian. Γ ¯ e acts transitively on Y . (b) (F, νφ ) is ergodic if and only if Γ ¯ (F, νφ ) is mixing if and only if Γt acts transitively on Y . A quantitative version of this result was obtained in [4]. We say that a set S ⊂ G is Diophantine on Y if there exist constants K and σ such that, for any function h on Y with ∆h = λh, there is an s ∈ S such that K khkL2 . λσ Let Γt (x, R) (Γe (x, R)) denote the set of g(P ) for all chains P = (x1 , x2 , . . . , xn ) with x1 = xn = x, n ≤ R, and dW ∗ (xj , xj+1 ) ≤ R (if xj+1 = f m xj , we require that |m| ≤ R). kh − h ◦ sk ≤
Proposition 8 [4]. (a) S is Diophantine on Y if and only if S is Diophantine on Y /[G, G] and Y /Center(G). (b) S is Diophantine on Y /Center(G) if and only if there exist no S-invariant functions, or, equivalently, S contains a finite Diophantine subset. (c) F is rapidly mixing if and only if Γt (R) is Diophantine for large R. It was proven in [1] that there is an open dense subset of pairs (f, τ ) such that Γt (R) = G for large R. The goal of this appendix is to prove the following statement. Theorem 3. Suppose that F is ergodic. If Γe (R) is Diophantine for large R, then any solution to (1) satisfies the tame estimates (2). Remark. Apparently, the above condition is also necessary for (2), but the approach of Section 4.4 (see also [4, Section 4.3]) shows only that, if Γe (R) is not Diophantine for large R and A = B − B ◦ F , then the norm of ∂yα B cannot be bounded by the norms of ∂yβ A. It does not eliminate the possibility that it is bounded by the norms of ∂yβ1 ∂xβ2 A, although this is unlikely.
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D. DOLGOPYAT
Proof. Note that the only place where we have used rapid mixing (i. e., Diophantineness of Γt (R)) was (10). Hence we need to show that (10) holds under the weaker condition that Γe (R) is Diophantine. To this end, we estimate (1 − Lλ )−1 , using the series −1 j ∞ 1 X 1 + Lλ 1 1 + Lλ −1 = . (1 − Lλ ) = 1− 2 2 2 j=0 2 Thus, instead of Proposition 1, we must prove the existence of C and s such that
n
1 + Lλ n
≤ Cλs 1 − 1
. (14)
2 Cλs The proof of (14) is similar to that of (7) which is Proposition 4.4 of [4]. Let us describe the modifications needed. Repeating the arguments from [4, p. 184], we can show that, if (14) fails, then for each C1 and β4 there exist a λ and an H λ m(λ) Hk ≥ 1 − |λ|−β4 , where such that kHkC 0 ≤ 1, L(H) ≤ Constλ, and k( 1+L 2 ) m(λ) = C1 ln λ and L(H) denotes the Lipschitz norm H : Σ+ → L2 (Y ). As in [4], this implies that kπλ (τm (˜ ω ))H(˜ ω ) − πλ (τm (ˆ ω ))H(ˆ ω )k ≤ λ−β5
for any ω ˜ and ω ˆ,
where β5 → ∞ as β4 → ∞. However, in the present setting, we also have kπλ (τ (ω))H(ω) − H(σω)k ≤ λ−β5 m λ Indeed, the expression for 1+L H (σω) 2
for all ω.
(15)
(1/2)m [H(σω) + eφ(ω) πλ (τ (ω))H(ω)] among the other terms. These two vectors should be almost collinear in the sense of [4, p. 185], which proves (15). Inequality (15) implies that, in our setting, Lemma 4.7 of [4] holds not only for t-chains, as in [4], but also for e-chains. Continuing as in [4, p. 186], we show that if (14) is false, then Γe cannot ne Diophantine. Thus, (14) holds. This proves Theorem 1 under the assumption that Γe (R) is Diophantine. References [1] M. I. Brin, Toplogical transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funktsional. Anal. i Prilozhen. 9 (1975), no. 1, 9–19 (Russian). MR 0370660. English translation: Funct. Anal. Appl. 9 (1975), 8–16. [2] M. I. Brin, The topology of group extensions of C-systems, Mat. Zametki 18 (1975), no. 3, 453–465 (Russian). MR 0394764 [3] R. de la Llave, J. M. Marco, and R. Moriy´ on, Canonical perturbation theory of Anosov systems and regularity results for the Livˇsic cohomology equation, Ann. of Math. (2) 123 (1986), no. 3, 537–611. MR 0840722 [4] D. Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157–205. MR 1919377 [5] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), no. 2, 389–449. MR 2031432
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[6] M. Field, I. Melbourne, and A. T¨ or¨ ok, Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions, Ergodic Theory Dynam. Systems 23 (2003), no. 1, 87–110. MR 1971198 [7] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2) 146 (1997), no. 2, 295–344. MR 1477760 [8] A. S. Gorodetskii and Y. S. Ilyashenko, Some properties of skew products over a horseshoe and a solenoid, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 96–118 (Russian). MR 1841753. English translation: Proc. Steklov Inst. Math. 231 (2000), no. 4, 90–112. [9] S. Hurder and A. Katok, Differentiability, rigidity and Godbillon–Vey classes for Anosov ´ flows, Inst. Hautes Etudes Sci. Publ. Math. (1990), no. 72, 5–61 (1991). MR 1087392 [10] J.-L. Journ´ e, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana 4 (1988), no. 2, 187–193. MR 1028737 [11] A. Katok and A. Kononenko, Cocycles’ stability for partially hyperbolic systems, Math. Res. Lett. 3 (1996), no. 2, 191–210. MR 1386840 [12] A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher rank abelian ´ groups and applications to rigidity, Inst. Hautes Etudes Sci. Publ. Math. (1994), no. 79, 131–156. MR 1307298 [13] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Ast´ erisque (1990), no. 187–188, 268. MR 1085356 [14] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), no. 3, 495–508. MR 1738057 [15] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 449–473. MR 0863205 University of Maryland, Department of Mathematics, 4417 Mathematics Bldg, College Park MD 20742 USA E-mail address: [email protected]
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC OF HYPERBOLIC SYSTEMS DMITRY DOLGOPYAT Dedicated to Yu. S. Ilyashenko on the occasion of his 60th birthday
Abstract. We prove Livsiˆc-type theorems for rapidly mixing compact group extensions of Anosov diffeomorphisms. 2000 Math. Subj. Class. 3730C, 37D30, 37J40. Key words and phrases. Cocycle equation, transfer operator, partial hyperbolicity, small divisors.
1. Introduction In recent years, several new phenomena in dynamics were discovered by looking at small perturbations of compact group extensions of hyperbolic systems [8], [14]. In view of this, it is desirable to develop a general theory of perturbations of such systems. The first step towards this goal is to understand infinitesimal perturbations, that is, study homological equations over such systems. In this paper, we study the regularity of solutions to cocycle equations. Regularity theory plays an important role in rigidity theory. Two of the most studied cases are translations of Td and Anosov diffeomorphisms (see [7], [11], [15] for the analysis of some other systems). The systems considered in this paper exhibit a mixture of hyperbolic and elliptic behaviors. Let M be a compact C ∞ Riemannian manifold, and let f : M → M be an Anosov diffeomorphism. Suppose that G is a compact connected Lie group, H a Lie subgroup of G, Y = G/H, and τ ∈ C ∞ (M, G). Let N = M × Y . We define F : N → N by F (x, y) = (f x, τ (x)y). A function A on N is called a coboundary if A = B − B ◦ F.
(1)
If B is bounded, H¨ older, smooth, etc., we say that the coboundary A is bounded, H¨older, smooth, etc. Take φ ∈ C α (M ); let µφ be the Gibbs state with potential φ, and let dνφ = dµφ dy. In this paper, we prove the following theorem. Take z0 ∈ N . Received May 14, 2003. c
2005 Independent University of Moscow
55
56
D. DOLGOPYAT
Theorem 1. Let F be rapidly mixing. If A ∈ C ∞ (N ) is a coboundary in L2 (νφ ) for some H¨ older function φ, then A is a C ∞ coboundary. In particular, if A is a bounded coboundary, then it is C ∞ . Moreover, there exists a k0 such that, if A belongs to C k (N ), then B belongs to C k−k0 (N ). If B satisfies the normalization condition B(z0 ) = 0, then kBkk−k0 ≤ ConstkAkk . (2) We refer the reader to the next section for the definition of rapid mixing. Recall [4] that a generic extension is rapidly mixing. Observe that Theorem 1 implies, in particular, that the set of coboundaries is closed in C k for k > k0 . We also present versions of this theorem for extensions of subshifts ofR finite type. Our results are also true for relative coboundaries. Let A0 (x) = A(x, y)dy. We say that A is a relative coboundary if A = A0 + B − B ◦ F.
(3)
Theorem 2. The assertions of Theorem 1 are valid for relative coboundaries. 2. Preliminaries 2.1. Subshifts of finite type. Here we present some results about subshifts of finite type and their compact extensions. The proofs can be found in [13, Chapters 3 and 8]. For a geometric interpretation of the results about extensions, see, e. g., [4, Section 2]. Let a be a finite alphabet; by A we denote a Card(a) × Card(a)-matrix whose entries are zeroes and ones. Suppose that Σ = ΣA is the associated (twosided) subshift of finite type, that is, Σ = {{ωi }+∞ i=−∞ : ωi ∈ a and Aωi ,ωi+1 = 1}. Let σ be the shift defined by σ(ω)i = ωi+1 . Given θ < 1, we consider the metric dθ on Σ given by dθ (ω 0 , ω 00 ) = θj , where j = max(k : ωi0 = ωi00 for |i| < k). Let Cθ (Σ) denote the set of dθ -Lipschitz functions. Given φ ∈ Cθ (Σ), we denote the Gibbs measure with potential φ by µφ , that is, hµφ + µφ (φ) = sup(hµ + µ(φ)), µ
where the supremum is over all σ-invariant probability measures. We shall use the fact that homologous functions have the same Gibbs measures. Let G be a ˜ = Σ × Y and compact connected Lie group, and let τ ∈ Cθ (Σ, G). We set N ˜ ˜ ˜ define F : N → N by F (ω, y) = (σω, τ (x)y). Consider the measure νφ given by ˜ ) = Cθ (Σ, C k (G)). We say that F˜ is rapidly mixing if, dνφ = dµφ dg. Let Ck,θ (N for any φ and N , there exists a k such that νφ (A(ω, y)B(F˜ n (ω, y))) − νφ (A)νφ (B) ≤ ConstkAkk,θ kBkk,θ n−N for all A, B ∈ Ck,θ . It was shown in [4, Theorem 4.3] that rapid mixing is generic among compact group extensions of subshifts of finite type. Let Σ+ be the associated one-sided subshift, which is defined similarly to Σ + but for the one-sided sequence ω = {ωi }∞ i=0 . The set Cθ (Σ ), Gibbs states, rapid
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC
57
mixing, etc. are defined for one-sided shifts as for two-sided shifts. Suppose that ˜ →N ˜ is the skew extension defined by τ ∈ Cθ (Σ, G), φ ∈ Cθ (Σ) is a potential, F˜ : N and A ∈ Ck,θ is an observable. Then there are τ ∗ ∈ C√θ (Σ, G), M ∈ C√θ (Σ, G), φ∗ ∈ Cθ (Σ+ ), ψ ∈ C√θ (Σ), A∗ ∈ Ck,√θ (Σ+ ), and K ∈ Ck,√θ (Σ) such that τ ∗ = (M ◦ σ)τ M −1 ,
φ∗ = φ + ψ − (ψ ◦ σ),
A∗ = A + K − K ◦ F˜ .
Moreover, φ∗ can be chosen so that X ∗ eφ ($) = 1 for any ω.
(4)
σ$=ω
The skew products defined by τ and τ ∗ are conjugate, φ and φ∗ have the same Gibbs measure, and A is a coboundary if and only if A∗ is a coboundary. ˜ + = Σ+ × Y . Suppose that F˜ is the skew extension determined by some Let N τ ∈ Cθ (Σ+ , G), ∆ is a G-invariant Laplacian on G, and Hλ = {ϕ : ∆ϕ = λϕ}. We endow Hλ with the L2 -norm. We set Cλ,θ (Σ+ ) = Cθ (Σ+ , Hλ ). Let φ be any H¨older function on Σ+ such that X eφ($) = 1 for any ω, (5) σ$=ω
and let µφ be the Gibbs measure for φ. Consider the transfer operator X (L(h))(ω, g) = eφ($) h($, τ −1 ($)g).
(6)
σ$=ω
It preserves Cλ,θ (Σ+ ). Let Lλ denote the restriction of L to Cλ,θ (Σ+ ). Proposition 1 [4, Proposition 4.4]. If F˜ is rapidly mixing, then there exist a C and an s such that n 1 n s kLλ k ≤ Cλ 1 − . (7) Cλs 2.2. Anosov diffeomorphisms. Recall that a diffeomorphism F : M → M is said to be Anosov if there exist an f -invariant splitting T M = Es ⊕ Eu and constants C, ρ < 1 such that kdf n vk ≤ Cρn kvk for any v ∈ E s
and kdf −n vk ≤ Cρn kvk for any v ∈ E u .
The distributions E s and E u are uniquely integrable, they are tangent to the foliations W s and W u , respectively. Since W s and W u are transverse, if x, y ∈ M u s are close to each other, then the intersection Wloc (x) ∩ Wloc (y) consists of one point, which we denote by [x, y]. A set Π is called a parallelogram if [x, y] ∈ Π for all x, y ∈ Π. A partition Π = {Π1 , Π2 , . . . Πn } is said to be Markov if, for all x ∈ Int(Πi ), f WΠs (x) ∈ WΠs (f x), f −1 WΠu (x) ∈ WΠu (f −1 x),
58
D. DOLGOPYAT
∗ where WΠ∗ (z) = Wloc (z) ∩ Πj for z ∈ Πj . Given a Markov partition Π, we can consider the subshift of finite type Σ for which a = {1, 2 . . . n} and Aij = 1 if and only if f (IntPi ) ∩ Pj = 0. The map ζ : Σ → M given by \ ζ(ω) = f −j Πωj j
defines a semicongugacy between σ and f . For a function τ from M to G, let τ¯ = τ ζ. Then ζ × id is a semicongugacy between F (x, y) = (f x, τ (x)y) and F˜ (ω, y) = (σω, τ¯(ω)y). We shall use the fact that the skew extension F is partially hyperbolic. That is, there exist an F -invariant splitting T N = EFs ⊕ EFc ⊕ EFu and constants C, ρ < 1 such that kdF n vk ≤ Cρn kvk for any v ∈ EFs ,
kdF −n vk ≤ Cρn kvk for any v ∈ EFu
and EFc is the tangent space to the fibers. The definition of Gibbs states for f is similar to that for σ. An important special case is the so-called SRB measure, which is the Gibbs measure with potential φSRB = − ln det(df |E u ). The importance of the SRB measure comes from the fact that, if Φ ∈ C(M ), then n−1 1X Φ(f j x) → µSRB (Φ) n j=0
as n → +∞
for Lebesgue almost all x. For any α, there exists θ such that if φ ∈ C α (M ), then φ¯ = φ ◦ ζ ∈ Cθ (Σ); µφ is a Gibbs state for f if and only if µφ¯(Ω) = µφ (ζ(Ω))
for any Ω ⊂ Σ.
We say that F is rapidly mixing if for any φ and N , there exists k such that νφ (A(x, y)B(F n (x, y))) − νφ (A)νφ (B) ≤ ConstkAkk kBkk n−N for any A, B ∈ C k (M ). The extension F is rapidly mixing if and only if F˜ is rapidly mixing. 3. Symbolic Systems ˜+ → N ˜ + is a rapidly mixing 3.1. One-sided shifts. In this subsection, F˜ : N + + extension of the one-sided subshift. Let Cr,θ (Σ ) = Cθ (Σ , C r (Y )). Lemma 1. Let A ∈ C∞,θ (Σ+ ) be an L2 (νφ )-coboundary for some φ ∈ Cθ (Σ+ ) such that A = B − B ◦ F˜ , where B ∈ L2 (νφ ). Then B has a version in C∞,θ (Σ+ ). Moreover, there exists a k0 such that, if A ∈ Ck,θ (Σ+ ), then B ∈ Ck−k0 ,θ (Σ+ ) and kBkk−k0 ,θ ≤ Const(k)kAkk,θ .
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC
59
Proof. According to what R we can assume that φ satisfies P was said in Section 2.1, (5). Let A = A0 + λ6=0 Aλ , where A0 (ω) = A(ω, g)dg and Aλ ∈ Hλ . Let P B = λ Bλ . Since F commutes with projections to Hλ , we have Aλ = Bλ − Bλ ◦ F˜ .
(8) +
In particular, A0 = B0 − B0 ◦ σ and, according to [13], B0 ∈ Cθ (Σ ). Hence we can assume without loss of generality that A0 ≡ 0. Applying Lλ to (8), we obtain Lλ Aλ = (Lλ − 1) Bλ . Thus, Bλ = −(1 − Lλ )−1 Lλ Bλ . There exists a p = p(G) such that Const kAλ kλ ≤ k/2−p kAkk,θ . λ By Proposition 1, there exists an s such that k(1 − Lλ )−1 k ≤ Const λs . Hence kBλ kλ ≤ Const λ2s kAλ kλ ≤
Const λk/2−(2s+p)
We have ¯ kBλ kk−k0 ,θ ≤ Const λp+
Let B =
k−k0 2
(9)
(10) kAkk,θ .
kBλ kλ .
(11) (12)
P
Bλ . Then X X ¯ 0 /2) kBkk−k0 ,θ ≤ kBλ kk−k0 ,θ ≤ Const λp+p+2s−(k kAkk,θ , λ
λ
λ
and this series converges if k0 is large enough. This completes the proof.
˜ →N ˜ be an extension of the two-sided subshift 3.2. Two-sided shifts. Let F˜ : N of finite type. Lemma 2. Let A = B − B ◦ F˜ . If A ∈ C∞,θ (Σ), then B ∈ C∞,θ1/4 (Σ). Moreover, there exists a k0 such that, if A ∈ Ck,θ (Σ), then B ∈ Ck−k0 ,θ1/4 (Σ) and kBkk−k0 ,θ1/4 ≤ Const(k)kAkk,θ . Proof. Let τ ∗ = (M ◦ σ)τ M −1 . Then the change of variables y ∗ = M y conjugates F˜ and F ∗ (ω, y ∗ ) = (σω, τ ∗ (ω)y ∗ ). Thus, A is an F˜ -coboundary if and only if A∗ (ω, y ∗ ) = A(ω, M −1 y ∗ ) is an F ∗ -coboundary. Let us represent A∗ as A∗ = A∗∗ +K ∗ −K ∗ ◦F ∗ , where A∗∗ ∈ Ck,θ1/4 (Σ+ ). Then A∗ is an F ∗ -coboundary if and only if A∗∗ is. But by Lemma 1, A∗∗ = B ∗∗ − B ∗∗ ◦ F , where B ∗∗ ∈ Ck−k0 ,θ1/4 (Σ+ ). Thus, A = (B + K) − (B + k) ◦ F˜ , where B(ω, y) = B ∗∗ (ω, y) and K(ω, y) = K ∗ (ω, M y). This proves the lemma. Corollary 1. If ω is a periodic orbit of σ, say σ n ω = ω, then n−1 X j Aλ (F˜ (ω, y)) ≤ Cλs kAλ kθ,k0 d(τn (ω)y, y). j=0
60
D. DOLGOPYAT
Proof. We have
n−1 X √ j Aλ (F˜ (ω, y)) ≤ |Bλ (ω, τn (ω)y) − Bλ (ω, y)| ≤ λ d(τn (ω)y, y)kBλ k, j=0
and the required assertion follows from (11).
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4. Anosov Diffeomorphisms 4.1. H¨ older continuity. We proceed to prove Theorem 1. In this section, we shall show that B has a H¨ older version. Take a Markov partition Π of M . Let Σ be the associated subshift of finite type, and let ζ : Σ → M be the semiconjugacy ζ ◦ σ = f ◦ ζ. ¯ −B ¯ ◦ F˜ , and We set A¯ = A ◦ ζ, A¯ = B Consider τ¯ and F˜ defined as in Section 2.2. P −1 ¯=P B ¯λ . Let Bλ = B ¯λ ◦ ζ −1 and B = B B . Since ζ is discontinuous, we λ λ λ cannot assert that the Bλ are H¨older, but we can state the following lemma, which is a consequence of the periodic leaf estimates of Corollary 1. Suppose p = (x, y) has a dense orbit. We have n−1 X Bλ (F n p) = Bλ (p) − Aλ (F j p). j=0
Lemma 3. Bλ |Orb(x,y) is uniformly H¨ older continuous with H¨ older constant CkAλ kλs . Proof. Suppose that m < n and d(F n p, F m p) ≤ ε. We set k = n − m, z = f m x, and q = F m p. By the Anosov closing lemma, there exists an x ˜ ∈ M such that f k x ˜=x ˜ and d(f j z, f j y) ≤ Cd(f k z, z)γ ρmax(j,k−j) for some γ > 0 and ρ < 1. Let u = (˜ x, y). Then k−1 k−1 k−1 X X X j j j j k Bλ (f q) − Bλ (q) = Aλ (F u) + [Aλ (F q) − Aλ (F u)] . Aλ (F q) ≤ j=0
j=0
j=0
By Corollary 1, the first part is O(kAλ kλs4 dα (p, F k p)) and the second part is
√ O(kAλ k λdγ (p, F k p)), √ because Aλ is Lipschitz with constant λkAλ k.
Since Orb(p) is dense, we can extend the Bλ to H¨older functions on N . Lemma 4. Under the conditions of Theorem 1, the restriction of B to each fiber is smooth. Moreover, there exists a k0 such that kBkC α0 (M,C k−k0 (G)) ≤ Const(k)kAkC α (M,C k (G)) . Proof. First, let us show that B has a H¨older version. By Lemma 3, each Bλ admits an extension from Orb(x, g) to N which is H¨older with H¨older norm at most ConstkAλ kλs . By the continuity of Aλ , this extension satisfies Aλ = Bλ − Bλ ◦ F , and Const kAλ k ≤ (k/2)−p kAkC k (N ) . λ
62
D. DOLGOPYAT
For B =
P
λ
Bλ , we have !
kBkC α (N ) ≤
X
kBλ kC α (N ) ≤ Const
λ
X
λ
p+s−k/2
kAkC k (N ) ,
λ
and this series converges if k is large enough. In other words, there exists a k1 such that BC α (N ) ≤ ConstkAkC k1 N . Applying this to ∆m A, we obtain kBkC α (M,H2m (G)) ≤ Constk∆m BkC α (N ) ≤ Constk∆m AkC k1 (N ) ≤ ConstkAkC k1 +2m (N ) and the application of the Sobolev embedding theorem completes the proof.
4.2. Smoothness. In this section, we prove the smoothness of B in the transverse directions. Lemma 5. Restrictions of B to the leaves of WFs and WFu are smooth. Proof. It suffices to consider WFs . We have A(p) = B(p) − B(F p). Thus, B(p) = A(p) + B(F p). Hence if p ∈ W s (p0 ) then B(p) − B(p0 ) =
∞ X
[A(F p j) − A(F j p0 )].
j=0
Since F j are contractions on WFS , this series can be differentiated term by term arbitrarily many times (see [3]). Now, we need the following fact [10]. Proposition 2 (Journe lemma). Let F1 and F2 be two continuous transverse foliations with smooth leaves. If B is a continuous function whose restrictions to the leaves of F1 and F2 are smooth, then B is smooth. Moreover, there exists a k0 such that, if the restrictions of B to the leaves are C k , then B is C k−k0 . This proposition, together with Lemmas 4 and 5, implies that B is smooth on each leave of W sc ; since it is also smooth on each leave of W u , we conclude that B is smooth. This completes the proof of Theorem 1. Remark. The weaker versions of the Journe lemma proven in [3], [9] are sufficient for the proof. 4.3. Relative coboundaries. Proof of Theorem 2. Apply Theorem 1 to ∆A. 4.4. A counter-example. Take G = T2 and F (x, t1 , t2 ) = (f x, t1 + α1 r(x), t2 + α2 r(x)). Suppose that α1 /α2 is irrational and, for all N , there exist m1,N , m2,N ∈ Z such that |α1 m1,N + α2 m2,N | ≤ m−N 2,n .
ˆ THEORY FOR COMPACT GROUP EXTENSIONS LIVSIC
Changing the indexation if necessary, we can assume that m2,N > N 2 . ΦN (x, t1 , t2 ) = exp(2πi(m1,N t1 + m2,N t2 )). Then
63
Let
Φ ◦ F = exp(2πi(m1,N α1 + m2,N α2 )r(x))ΦN . Let AP= N ((ΦN − Φ ◦ F )/N 2 ). Then A ∈ C ∞ (N ) and A = B − B ◦ F m where B = N ΦN ∈ C 0 (N ) − C 1 (N ). Considering suitable linear combinations of ΦN , we see that F is not rapidly mixing. This shows that Theorem 1 is not valid for arbitrary extensions. P
4.5. Obstructions. In this section, we give criteria for a function to be a coboundary. Most of these criteria come from other papers, however their applicability is a consequence of the fact that different notions of coboundaries coincide in the situation under consideration. Sometimes, it is easier to verify that A is a relative coboundary. It is sufficient, because it is well known when A0 is an f -coboundary. (i) We set Dφ (A) = νφ (A2 ) − νφ2 (A) + 2
∞ X
[νφ (A(A ◦ F˜ j ))νφ2 (A)].
j=1
Proposition 3 [6]. A is cohomologous to a constant if and only if there exists a φ such that Dφ (A) = 0, or, equivalently, Dφ (A) = 0 for any ψ. Proof. Without loss of generality, we can assume that νφ (A) = 0. Then Dφ (A) = 0 if an only if A is an L2 (νφ )-coboundary (by the spectral theorem). The latter condition is equivalent to A being a H¨older coboundary (by Theorem 1), which is, in turn, equivalent to A being an L2 (νψ )-coboundary for each ψ. (ii) Let P = {p0 , p1 . . . pn } be a chain such that pk+1 ∈ W s (pk ) ∪ W u (pk ). We say that P is closed if p0 = pn . We set X r(P ) = r(pk , pk+1 ), k
where
r(pk , pk+1 ) =
∞ X A(F j pk+1 ) − A(F j pk )
if pk+1 ∈ W s pk ,
j=0
−1 X A(F j pk ) − A(F j pk+1 ) if pk+1 ∈ W s pk . j=−∞
The following statement is Corollary 3.1 from [11]. Proposition 4. If F has the accessibility property, then A is cohomologous to a constant if and only if r(P ) = 0 for any closed chain P . (iii) The next result follows from the proof of Theorem 1. Proposition 5. A is a relative coboundary if and only if ∆N A is a coboundary for any N ∈ N.
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D. DOLGOPYAT
(iv) Let G = T, and let ν be the SRB measure for F . Consider the one-parameter family Fε (x, z) = (f x, z + τ (x) + εA(x, z) + ε2 α(ε, x, z). Suppose that νε is a u-Gibbs measure for Fε , that is, the projection of νε to M is µSRB . Proposition 6. A is a relative coboundary if and only if λc (νε ) = o(ε2 ). Proof. We use the asymptotics [5] νε (H) = ν(H) + εω(H) + o(ε), where ω(H) =
∞ X dH . ν A ◦ F −j dz j=1
ε We want to apply this to Hε = ln dF dz . We have
So
dFε dA dα(0, x, z) =1+ε + ε2 + o(ε2 ). dz dz dz " 2 # dA 1 dA dFε 2 dα(0, x, z) =ε +ε − ln + o(ε2 ). dz dz dz 2 dz
Hence dA dFε νε ln = εν dz dz " 2 X # ∞ 2 1 dA dα(0, x, z) −j d A 2 ν A◦F − ν + + o(ε2 ). +ε ν 2 dz 2 dz dz j=1 Since dν = dµSRB dg, it follows that dα(0, x, z) dA =ν = 0 and ν dz dz d2 A dA dA ν A ◦ f −j 2 = −ν ◦ F −j . dz dz dz Hence " 2 X # ∞ dA dA dFε 2 1 −j dA ∼ −ε ν − ν ◦F , νε ln dz 2 dz dz dz j=1 that is, ε2 DSRB ( dA dFε dz ) νε ln ∼− . dz 2
(13)
Therefore, λc (νε ) = o(ε2 ) dA if and only if DSRB ( dA dz ) = 0. By Proposition 3, dz is a coboundary, which means that A satisfies (3).
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Appendix A. The Non-Mixing Case Note that Theorem 1 may be valid for an Anosov time rotation even though it is not mixing. In this appendix, we give an extension of Theorem 1 to the nonmixing case. In order to explain the result, we recall some background. Given ¯ = M × G and consider the principal extension F¯ : N ¯ → N ¯ given by τ , let N ¯ F (x, g) = (f (x), τ (x)g). Recall the definition of Brin groups [1], [2]. Given a partially hyperbolic diffeomorphism, we call a sequence P = {p1 , p2 , . . . , pn } a e-chain (respectively t-chain) if pj+1 ∈ W u (pj ) ∪ W s (pj ) (respectively, pj+1 ∈ W u (pj ) ∪ W s (pj ) ∪ Orb(pj )). Take a reference point x ∈ M . Given any chain ¯ starting at P ⊂ M with xn = x1 = x and any g1 ∈ G, there is unique chain P¯ ⊂ N ¯ (x, g1 ) and covering P . P is not closed, instead we have gn = g(P )g1 . Let Γt (x) (Γe (x)) denote the set of all g(P ) for all closed t-chains (respectively, e-chains) starting at x. Proposition 7 (Brin, [1], [2]). (a) The Γ∗ (x) are groups. The Γ∗ of different points are conjugate, Γt is a normal subgroup of Γe , and Γe /Γt is cyclic. In particular, ¯ e /Γ ¯ t is Abelian. Γ ¯ e acts transitively on Y . (b) (F, νφ ) is ergodic if and only if Γ ¯ (F, νφ ) is mixing if and only if Γt acts transitively on Y . A quantitative version of this result was obtained in [4]. We say that a set S ⊂ G is Diophantine on Y if there exist constants K and σ such that, for any function h on Y with ∆h = λh, there is an s ∈ S such that K khkL2 . λσ Let Γt (x, R) (Γe (x, R)) denote the set of g(P ) for all chains P = (x1 , x2 , . . . , xn ) with x1 = xn = x, n ≤ R, and dW ∗ (xj , xj+1 ) ≤ R (if xj+1 = f m xj , we require that |m| ≤ R). kh − h ◦ sk ≤
Proposition 8 [4]. (a) S is Diophantine on Y if and only if S is Diophantine on Y /[G, G] and Y /Center(G). (b) S is Diophantine on Y /Center(G) if and only if there exist no S-invariant functions, or, equivalently, S contains a finite Diophantine subset. (c) F is rapidly mixing if and only if Γt (R) is Diophantine for large R. It was proven in [1] that there is an open dense subset of pairs (f, τ ) such that Γt (R) = G for large R. The goal of this appendix is to prove the following statement. Theorem 3. Suppose that F is ergodic. If Γe (R) is Diophantine for large R, then any solution to (1) satisfies the tame estimates (2). Remark. Apparently, the above condition is also necessary for (2), but the approach of Section 4.4 (see also [4, Section 4.3]) shows only that, if Γe (R) is not Diophantine for large R and A = B − B ◦ F , then the norm of ∂yα B cannot be bounded by the norms of ∂yβ A. It does not eliminate the possibility that it is bounded by the norms of ∂yβ1 ∂xβ2 A, although this is unlikely.
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D. DOLGOPYAT
Proof. Note that the only place where we have used rapid mixing (i. e., Diophantineness of Γt (R)) was (10). Hence we need to show that (10) holds under the weaker condition that Γe (R) is Diophantine. To this end, we estimate (1 − Lλ )−1 , using the series −1 j ∞ 1 X 1 + Lλ 1 1 + Lλ −1 = . (1 − Lλ ) = 1− 2 2 2 j=0 2 Thus, instead of Proposition 1, we must prove the existence of C and s such that
n
1 + Lλ n
≤ Cλs 1 − 1
. (14)
2 Cλs The proof of (14) is similar to that of (7) which is Proposition 4.4 of [4]. Let us describe the modifications needed. Repeating the arguments from [4, p. 184], we can show that, if (14) fails, then for each C1 and β4 there exist a λ and an H λ m(λ) Hk ≥ 1 − |λ|−β4 , where such that kHkC 0 ≤ 1, L(H) ≤ Constλ, and k( 1+L 2 ) m(λ) = C1 ln λ and L(H) denotes the Lipschitz norm H : Σ+ → L2 (Y ). As in [4], this implies that kπλ (τm (˜ ω ))H(˜ ω ) − πλ (τm (ˆ ω ))H(ˆ ω )k ≤ λ−β5
for any ω ˜ and ω ˆ,
where β5 → ∞ as β4 → ∞. However, in the present setting, we also have kπλ (τ (ω))H(ω) − H(σω)k ≤ λ−β5 m λ Indeed, the expression for 1+L H (σω) 2
for all ω.
(15)
(1/2)m [H(σω) + eφ(ω) πλ (τ (ω))H(ω)] among the other terms. These two vectors should be almost collinear in the sense of [4, p. 185], which proves (15). Inequality (15) implies that, in our setting, Lemma 4.7 of [4] holds not only for t-chains, as in [4], but also for e-chains. Continuing as in [4, p. 186], we show that if (14) is false, then Γe cannot ne Diophantine. Thus, (14) holds. This proves Theorem 1 under the assumption that Γe (R) is Diophantine. References [1] M. I. Brin, Toplogical transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funktsional. Anal. i Prilozhen. 9 (1975), no. 1, 9–19 (Russian). MR 0370660. English translation: Funct. Anal. Appl. 9 (1975), 8–16. [2] M. I. Brin, The topology of group extensions of C-systems, Mat. Zametki 18 (1975), no. 3, 453–465 (Russian). MR 0394764 [3] R. de la Llave, J. M. Marco, and R. Moriy´ on, Canonical perturbation theory of Anosov systems and regularity results for the Livˇsic cohomology equation, Ann. of Math. (2) 123 (1986), no. 3, 537–611. MR 0840722 [4] D. Dolgopyat, On mixing properties of compact group extensions of hyperbolic systems, Israel J. Math. 130 (2002), 157–205. MR 1919377 [5] D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math. 155 (2004), no. 2, 389–449. MR 2031432
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