Lyapunov exponents of cocycles over non-uniformly hyperbolic systems

We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $\mu$. The cocycle $A$ over $f$ is Holder continuous and takes values in $GL(d,R)$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d,R)$-valued cocycle $A$ we prove that the Lyapunov exponents of $A$ with respect to $\mu$ can be approximated by the Lyapunov exponents of $A$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-dimensional setting one can define the upper and lower Lyapunov exponents of $A$ with respect to $\mu$, but they cannot always be approximated by the exponents of $A$ on periodic orbits. We prove that they can be approximated in terms of the norms of the return values of $A$ on hyperbolic periodic orbits of $f$.

1. Introduction and statements of the results. The theory of non-uniformly hyperbolic dynamical systems was pioneered by Ya. Pesin in [11,12] as a generalization of uniform hyperbolicity. It has become one of the central areas in smooth dynamics with numerous applications, see [2,13]. Periodic points play a major role in the study of both uniformly and non-uniformly hyperbolic systems. In the non-uniformly hyperbolic case, the existence of hyperbolic periodic orbits and their relations to dynamical and ergodic properties of the system were established by A. Katok in a seminal paper [9]. In fact, any hyperbolic invariant measure can be approximated in weak* topology by invariant measures supported on hyperbolic periodic points of the system [2]. A further advance in this direction was obtained by Z. Wang and W. Sun who showed that Lyapunov exponents of any hyperbolic measure can be approximated by Lyapunov exponents at periodic points [14]. This does not follow from weak* approximation as Lyapunov exponents in general do not depend continuously on the measure in weak* topology.
The Lyapunov exponents above correspond to the derivative cocycle Df of the base system (X, f ), which is a particular case of a linear cocycle, that is, an automorphism of a vector bundle over X that projects to f . Linear cocycles are the prime examples of non-commutative cocycles over dynamical systems. For uniformly hyperbolic systems, where every invariant measure is hyperbolic, a periodic 5106 BORIS KALININ AND VICTORIA SADOVSKAYA approximation of Lyapunov exponents of linear cocycles was established by the first author in [5]. The results and techniques in [5] proved useful in various areas such as cohomology of non-commutative cocycles and the study of random and Markovian matrices and operators. More recently, approximation results were obtained by the authors for cocycles with values in the group of invertible bounded linear operators on a Banach space [7] and by L. Backes for semi-invertible matrix cocycles [1].
In this paper we extend the periodic approximation results to linear cocycles over non-uniformly hyperbolic systems. In the base we consider a diffeomorphism f of a compact manifold X preserving a hyperbolic ergodic probability measure µ. The cocycles over (X, f ) will take values in GL(d, R) or, more generally, the group GL(V ) of invertible bounded linear operators on a Banach space V . The space L(V ) of bounded linear operators on V is a Banach space equipped with the operator norm A = sup { Av : v ∈ V, v ≤ 1}. The open set GL(V ) of invertible elements in L(V ) is a topological group and a complete metric space with respect to the metric Definition 1.1. Let f be a homeomorphism of a compact metric space X and let A be a function from X to GL(V ). The GL(V )-valued cocycle over f generated by A is the map A : X × Z → GL(V ) defined by A(x, 0) = Id and for n ∈ N, In the finite-dimensional case of V = R d we will call A a GL(d, R)-valued cocycle. We say that the cocycle A is α-Hölder, 0 < α ≤ 1, if there exists M > 0 such that Clearly, A satisfies the cocycle equation A n+k Since X is compact, the Hölder condition (1) is equivalent to A x −A y ≤ M dist(x, y) α for all x, y ∈ X. Hölder continuity of a cocycle is natural in our setting as the lowest regularity allowing development of a meaningful theory beyond the measurable case. It is also the regularity of the derivative cocycle of a C 1+Hölder diffeomorphism and its restrictions to Hölder continuous invariant sub-bundles.
Any GL(V )-valued cocycle A can be viewed as an automorphism of the trivial ). More generally, we can consider a linear cocycle A, i.e. an automorphism of any vector bundle E over X that projects to f . This setting covers the case of the derivative cocycle Df of a diffeomorphism f of X with nontrivial tangent bundle. For any measure µ on X, any vector bundle E over X is trivial on a set of full measure [2, Proposition 2.1.2] and hence any linear cocycle A can be viewed as a GL(V )-valued cocycle on a set of full measure.
First we consider the finite dimensional case where the Lyapunov exponents and Lyapunov decomposition for A with respect to an ergodic f -invariant measure µ are given by Oseledets Multiplicative Ergodic Theorem. We note that both are defined µ-a.e. and depend on the choice of µ.
Oseledets Multiplicative Ergodic Theorem. [10,2] Let f be an invertible ergodic measure-preserving transformation of a Lebesgue probability space (X, µ). Let A be a measurable GL(d, R)-valued cocycle over f satisfying log A x ∈ L 1 (X, µ) and log A −1 x ∈ L 1 (X, µ). Then there exist numbers λ 1 < · · · < λ m , an f -invariant set Λ with µ(Λ) = 1, and an A-invariant Lyapunov decomposition By Lyapunov exponents of A at a periodic point p = f k p we mean the Lyapunov exponents of A with respect to the invariant measure µ p on the orbit of p. They equal (1/k) of the logarithms of the absolute values of the eigenvalues of A k p . A periodic point p is called hyperbolic if Df has no zero exponents at p, that is D p f k has no eigenvalues of absolute value 1.
The following theorem extends the periodic approximation results in [14] and [5] to linear cocycles over non-uniformly hyperbolic systems. Theorem 1.4. Let f be a C 1+Hölder diffeomorphism of a compact manifold X, let µ be a hyperbolic ergodic f -invariant Borel probability measure on X, and let A be a GL(d, R)-valued Hölder continuous cocycle over f .
Then the Lyapunov exponents λ 1 ≤ · · · ≤ λ d of A with respect to µ, listed with multiplicities, can be approximated by the Lyapunov exponents of A at periodic points. More precisely, for each > 0 there exists a hyperbolic periodic point p ∈ X for which the Lyapunov exponents λ The largest and smallest Lyapunov exponents λ + (A, µ) = λ m and λ − (A, µ) = λ 1 can be expressed as follows: While there is no Multiplicative Ergodic Theorem in the infinite-dimensional case in general, the upper and lower Lyapunov exponents λ + and λ − of A can still be defined by (3), see Section 4.1. For the invariant measure µ p on the orbit of p = f k p we have In the infinite-dimensional setting, it is not always possible to approximate λ + (A, µ) by λ + (A, µ p ), even for cocycles over uniformly hyperbolic systems [7, Proposition 1.5]. However, an approximation of λ + (A, µ) by k −1 log A k p was obtained in [7] for cocycles over uniformly hyperbolic systems. The next theorem establishes such an approximation in the non-uniformly hyperbolic setting. Theorem 1.5. Let f be a C 1+Hölder diffeomorphism of compact manifold X, let µ be a hyperbolic ergodic f -invariant Borel probability measure on X, and let A be a Hölder continuous GL(V )-valued cocycle over f .

BORIS KALININ AND VICTORIA SADOVSKAYA
Then for each > 0 there exists a hyperbolic periodic point p = f k p in X such that The proof of the finite dimensional approximation in Theorem 1.4 relies on Multiplicative Ergodic Theorem, which yields that the cocycle has finitely many Lyapunov exponents and, in particular, the largest one is isolated. As this may not be the case in infinite dimensional setting even for a single operator, in Theorem 1.5 we use a different approach which relies on norm estimates. In particular, we use a suitable version of Lyapunov norm and results on subadditive cocycles [8].
Remark 1.6. Theorems 1.4 and 1.5 can be strengthened to conclude the existence of a hyperbolic periodic point p = f k p which gives simultaneous approximation as in (2) and (4) for finitely many cocycles A (i) , i = 1, . . . , m, over f with values in GL(V i ). We describe the modifications for this case in the proofs.
Remark 1.7. Theorems 1.4 and 1.5 hold if we replace X ×V by a Hölder continuous vector bundle E over X with fiber V and the cocycle A by an automorphism A : E → E covering f . This setting is described in detail in Section 2.2 of [6] and the proofs work without any significant modifications.

2.
Preliminaries. For a GL(V )-valued cocycle A over (X, f ) we consider the trivial bundle E = X × V and view A n x as a fiber map from E x to E f n x . This makes notations and arguments more intuitive and the extension to non-trivial bundles more transparent. In fact, all our arguments are written for the bundle setting, except we sometimes identify fibers E x and E y at nearby points x and y. This is automatic for a trivial bundle, and a detailed description of a suitable identification for a non-trivial bundle is given in [6, Section 2.2].
2.1. Lyapunov metric. We consider a GL(d, R)-valued cocycle A over (X, f, µ) as in the Oseledets Multiplicative Ergodic Theorem and denote the standard scalar product in R d by ·, · . We fix > 0 and for any point x ∈ Λ define the Lyapunov scalar product ·, · x, on R d as follows. For The series converges exponentially for any x ∈ Λ. The constant d in the formula allows a more convenient comparison with the standard scalar product. The norm generated by this scalar product is called the Lyapunov norm and is denoted by . x, . When is fixed we will denote the scalar product by ·, · x and the norm by . x . We summarize the main properties of the Lyapunov scalar product and norm, see [2, Sections 3.5.1-3.5.3] for more details. A direct calculation shows [2, Theorem 3.5.5] that for any x ∈ Λ and any u ∈ E i x , e nλ+− n ≤ A n x f n x←x ≤ e nλ++ n for all n ∈ N, where λ + = λ m is the largest Lyapunov exponent and . f n x←x is the operator norm with respect to the Lyapunov norms defined as The Lyapunov scalar product and norm are defined only on Λ and, in general, depend only measurably on the point even if the cocycle is Hölder, so comparison with the standard norm is important. The lower bound follows easily from the definition: u x, ≥ u . An upper bound is not uniform, but can be chosen to change slowly along the orbits [2, Proposition 3.5.8]: there exists a measurable function K (x) on Λ such that for all x ∈ Λ and u ∈ R d , and for all x ∈ Λ and n ∈ Z.
For any matrix A and any points x, y ∈ Λ inequalities (7) and (8) yield For any > 1 we define the sets and note that µ(Λ , ) → 1 as → ∞. Without loss of generality we can assume that the set Λ , is compact and that Lyapunov splitting and Lyapunov scalar product are continuous on Λ , . Indeed, by Luzin theorem we can always find a subset of Λ , satisfying these properties with arbitrarily small loss of measure.

2.2.
Pesin sets and closing lemma. Let f be a diffeomorphism of a compact manifold X and µ be an ergodic f -invariant Borel probability measure. We apply the Multiplicative Ergodic Theorem and construct the Lyapunov metric as above for the derivative cocycle A x = D x f . For this cocycle, we will denote the corresponding set Λ , defined in the previous section by R , , which is often called a Pesin set. Suppose now that the measure µ is hyperbolic, i.e. all Lyapunov exponents of the derivative cocycle Df with respect to µ are non-zero. We assume that there are both positive and negative such exponents. Otherwise µ is an atomic measure on a single periodic orbit [2,Lemma 15.4.2], in which case our results are trivial. We denote by χ > 0 the smallest absolute value for these exponents. We will fix > 0 sufficiently small compared to χ and ∈ N large enough so that the corresponding Pesin set R , has positive measure. We will apply the following closing lemma. It does not use the splitting into individual Lyapunov subbundles E i , only the stable/unstable sub-ones, which are the sums of all Lyapunov sub-bundles corresponding to negative/positive Lyapunov exponents, respectively. Consequently, a cruder version of a Pesin set can be used instead of R , .
While the lemma is usually stated with a constant on the right hand side of (11), this constant can be absorbed using the choice of β. We will not use hyperbolicity of the periodic point p in the proof. In fact, for sufficiently large k the hyperbolicity of p can be recovered by applying our argument to the derivative cocycle Df .
3. Proof of Theorem 1.4. Let λ 1 < · · · < λ m be the Lyapunov exponents of A with respect to µ, listed without multiplicities. We will denote the largest exponent λ m by λ and second largest λ m−1 by λ . Similarly, for any periodic point p we denote by λ (p) the largest Lyapunov exponent of A at p.
We fix > 0 sufficiently small compared to χ and ∈ N large enough so that the corresponding Pesin set R , for the derivative cocycle Df of the base system has positive measure. We apply the Closing Lemma 2.1 and get γ = γ( , ) > 0.
If λ is not the only Lyapunov exponent of A with respect to µ, we define and otherwise we set 0 = min { αγ, }. We fix 0 < < 0 and consider the sets Λ , for the cocycle A. We denote and fix sufficiently large so that µ(P ) > 0. We take a point x ∈ P which is in the support of µ restricted to P . Then for any β > 0 we have µ(P ∩ B β/2 (x)) > 0, where B β/2 (x) is the open ball of radius β/2 centered at x. By Poincare recurrence there are infinitely many k ∈ N such that f k x ∈ P ∩ B β/2 (x). For any such k we have: x, f k x ∈ P and dist(x, f k x) < β. Taking β = β(δ, , ) > 0 from the Closing Lemma 2.1 we obtain a hyperbolic periodic point satisfying (11). We can assume that β ≤ δ, so that when δ is small so is β.
We conclude that for each δ > 0 there exist arbitrarily large k such that x, f k x ∈ P and there is a hyperbolic periodic point p = f k p satisfying (11). Now we show that for such a point p with a sufficiently large k and a sufficiently small δ we have To estimate λ (p) from above we use the fact [5, Lemma 3.1] that for such a point p where the constant c depends only on A and on the number (αγ − ), which also follows from Lemma 4.3 below. Since we chose < αγ we obtain provided that δ < 1 and k is large enough compared to . Now we estimate λ (p) from below. We denote x i = f i x and p i = f i p, and we write . i for the Lyapunov norm at x i . Since the Lyapunov norm may not exist at points p i we will use the Lyapunov norms at the corresponding points x i for the estimates. For each i we have the orthogonal splitting xi is the Lyapunov space at x i corresponding to the largest Lyapunov exponent λ = λ m , and E i is the direct sum of the other Lyapunov spaces at x i . We will assume that λ is not the only Lyapunov exponent of A, as otherwise E i = {0} and the argument is simpler. For a vector u ∈ R d we write u = u 1 + u 2 , where u 1 ∈ E i and u 2 ∈ E i . We take θ = e λ −λ+4 < 1 by the choice of . For i = 0, . . . , k we consider the cones show that there exist δ 0 > 0 such that for all 0 < δ < δ 0 and all i = 0, . . . , k − 1 We fix 0 ≤ i < k and a vector u We Since both x 0 and x k are in Λ and dist(x i , p i ) ≤ δe −γ min{i, k−i} , using (9) we can estimate Since u i ≤ √ 2 u 1 i we conclude using (16) that Setting w = A pi u = (Id + ∆ i )A xi u = (Id + ∆ i )v we observe that and hence using (16) and (18) we obtain that for small enough δ which gives the inequality in (15). Similarly, using u 2 i ≤ u 1 i , we get for all sufficiently small δ. Finally, if u = 0 we get that This shows that w ∈ K θ i+1 and the inclusion A pi (K i ) ⊂ K θ i+1 in (15) follows. We conclude that (15) holds for each i = 0, . . . , k − 1 and hence A k p (K 0 ) ⊂ K θ k . Since Λ is chosen compact and so that the Lyapunov splitting and Lyapunov metric are continuous on it, the cones K θ 0 and K θ k are close if dist(x, f k x) < β is small. Thus we can ensure that K θ k ⊂ K 0 if β small enough and hence A k p (K 0 ) ⊂ K 0 . Finally, using the inequality in (15) for each i = 0, . . . , k − 1 we obtain that for any since Lyapunov norms at x and f k x are close if δ and hence β is small enough. Since A k p u ∈ K 0 for any u ∈ K 0 , we can iterate A k p and use the inequality above to estimate λ (p) from below by the exponent of any u ∈ K 0 : provided that k is large enough. This completes the proof of the approximation of the largest exponent (13).
To approximate all Lyapunov exponents of A we consider cocycles ∧ i A induced by A on the i-fold exterior powers ∧ i R d , for i = 1, . . . , d. The largest Lyapunov exponent of ∧ i A is (λ d + · · · + λ d−i+1 ), where λ 1 ≤ · · · ≤ λ d are the Lyapunov exponents of A listed with multiplicities. If a periodic point p = f k p satisfies then we obtain the approximation |λ i − λ with µ(P ) > 0, where Λ i are the corresponding sets for all cocycles ∧ i A, i = 1, . . . , d. Then the previous argument applies yielding (13) for each ∧ i A.
A similar argument shows that one can obtain a simultaneous approximation of all Lyapunov exponents for several cocycles.

Subadditive cocycles and their exponents.
A subadditive cocycle over a dynamical system (X, f ) is a sequence of measurable functions a n : X → R with a n+k (x) ≤ a k (x) + a n (f k x) for all x ∈ X and k, n ∈ N.
For any ergodic measure-preserving transformation f of a probability space (X, µ) and any subadditive cocycle over f with integrable a n , the Subadditive Ergodic Theorem yields that for µ almost all x lim n→∞ a n (x) n = lim n→∞ a n (µ) n = inf n∈N a n (µ) n =: ν(a, µ), where a n (µ) = X a n (x)dµ.
The limit ν(a, µ) ≥ −∞ is called the exponent of the cocycle a n with respect to µ. For a GL(V )-valued cocycle A over (X, f ) it is easy to see that a n (x) = log A n x andã n (x) = log (A n x ) −1 .
are subadditive cocycles over f . For these continuous cocycles the Subadditive Ergodic Theorem gives the existence of the limits in (3): for µ almost all x It is easy to see that λ − ≤ λ + and both are finite. Also, for µ almost all which implies that both equalities in (3) hold for x ∈ Λ. Clearly, µ(Λ) = 1. We will also use the following more detailed result on the behavior of subadditive cocycles established by A. Karlsson and G. Margulis.

Proposition 4.1. [8, Proposition 4.2]
Let a n (x) be an integrable subadditive cocycle with exponent λ > −∞ over an ergodic measure-preserving system (X, f, µ). Then there exists a set E ⊂ X with µ(E) = 1 such that for each x ∈ E and each > 0 there exists an integer L = L(x, ) so that the set S = S(x, , L) of integers n satisfying a n (x) − a n−i ( is infinite. (In fact, S has positive asymptotic upper density [3]).
We will use the following corollary of this result. n (x) be integrable subadditive cocycles with exponents λ (1) > −∞, . . . , λ (m) > −∞, respectively, over an ergodic measurepreserving system (X, f, µ). Then there exists a set E ⊂ X with µ(E) = 1 such that for each x ∈ E and each > 0 there exists an integer L = L(x, ) so that the set S = S(x, , L) of integers n satisfying the following condition is infinite: Proof. We apply Proposition 4.1 to the subadditive cocycle a n (x) = a (1) n (x) + · · · + a (m) n (x) with exponent λ = λ (1) + · · · + λ (m) > −∞ and obtain the set E for this cocycle. Then for each x ∈ E and each > 0 there exists L = L (x, ) and an infinite set S = S(x, , L ) such that for all n ∈ S we have m j=1 a (j) By the Subadditive Ergodic Theorem there exists a set G of full measure such that for each x ∈ G and each j = 1, . . . , m we have lim n→∞ n −1 a We set L = L(x, ) = max{L , M } and E = G ∩ E , with µ(E) = 1. Subtracting the inequalities above for j = 2, . . . , m from (27) we get that for all x ∈ E and n ∈ S The inequalities for the = cocycles a (j) n , j = 2, . . . , m, follow similarly.

4.2.
Lyapunov norm for upper and lower Lyapunov exponents. Since the Multiplicative Ergodic Theorem does not apply in the infinite dimensional setting, we use a cruder version of Lyapunov norm which takes into account only upper and lower Lyapunov exponents. We fix > 0 and for any point x ∈ Λ we define the Lyapunov norm of u ∈ E x as follows: By the definition (24) of Λ, both series converge exponentially. Properties of this norm were obtained in [7,Proposition 3.1]. They are similar to those of the usual Lyapunov norm discussed in Section 2.1. In particular, A n x xn←x0 ≤ e n(λ+ + ) and (A n x ) −1 x0←xn ≤ e n(−λ− + ) for x ∈ Λ (29) and the ratio to the background norm u x / u is "tempered". Specifically, there exist an f -invariant set Λ ⊂ Λ with µ(Λ ) = 1 and a measurable function K (x) on Λ satisfying conditions (8) and (7). For any > 1 we define and note that µ(Λ ) → 1 as → ∞.
We use this Lyapunov norm to obtain estimates similar to (29) for any point p ∈ X whose trajectory is close to that of a point x ∈ Λ . Since the Lyapunov norm may not exist at points f i p we will use the Lyapunov norms at the corresponding points f i x for the estimates.
We apply Corollary 4.2 to the subadditive cocycles a (1) n (x) = a n (x) = log A n x and a (2) n (x) =ã n (x) = log (A n x ) −1 and obtain the set E of full measure for these two cocycles.
We take a compact set K ⊂ (P ∩ E) with µ(K) > 0.7, let ν be the restriction of µ to K, and denote by G the support of ν. Then G is a compact subset of K satisfying ν(X \ G) = 0, ν(G) = µ(K) > 0.7, and ν(U ) > 0 for any subset U (relatively) open in G. We fix a countable basis {U j }, U j ⊂ G, for the topology of G and note that µ(U j ) = ν(U j ) > 0. We denote by G the subset of full µ-measure in G on which the Birkhoff Ergodic Theorem holds for the indicator functions of each U j , that is for each x ∈ G lim n→∞ n −1 {i : 0 ≤ i ≤ n − 1 and f i x ∈ U j } = µ(U j ) > 0 for all j ∈ N. (34) For the remainder of the proof we fix a point x ∈ G and choose σ = 4 /(αγ). We establish the following lemma, which is a refinement of [4, Lemma 8] for our setting.
Lemma 5.1. For each x ∈ G , β > 0 and σ > 0 there exists an integer N = N (x, σ, β) such that for each n ≥ N there is an integer k satisfying Proof. We fix x ∈ G and β > 0 and consider a set U j from the countable base that is contained in the open ball B(x, β/2). Since x ∈ G , condition (34) holds for x. Denoting the set of the return times i to U j by T and letting φ(n) = | T ∩ [0, n − 1] | we get that φ(n)/n → t = µ(U j ) > 0. For any σ > 0 we can take c > 0 such that and then M such that (1 − c)tn < φ(n) < (1 + c)tn for all n ≥ M.
Using this and the fact that (1 − c)(1 + 2σ) − (1 + c)(1 + σ) > 0 by the choice of c, we obtain that there exists N > M such that for all n ≥ N we have which means there exists k between n(1 + σ) and n(1 + 2σ) such that f k x is in U j ⊂ (G ∩ B(x, β/2)).
Since x ∈ G ⊂ E, by Corollary 4.2 there exists an integer L = L(x, ) such that for any n ∈ S x and any i with L ≤ i ≤ n a n (x) − a n−i ( We conclude that for any β > 0 there exist arbitrarily large n for which the above property holds and a corresponding k = k(n) satisfying the conclusion of Lemma 5.1. We will later choose δ > 0 sufficiently small so that (47) below is satisfied, and take β = β(δ, , γ) > 0 from the Closing Lemma 2.1, which then gives a periodic point p = f k p such that dist(f i x, f i p) ≤ δe −γ min{i, k−i} for every i = 0, . . . , k. (36) Obtaining upper estimates for A k p and (A k p ) −1 . Since x and p satisfy (36) we can apply Lemma 4.3 with i = k to get A k p ≤ e c δ α e k(λ+ + ) and (A k p ) −1 ≤ e c δ α e k(−λ− + ) .
For k sufficiently large compared to and c = c(A, αγ − ), we obtain k −1 log A k p ≤ λ + + + k −1 (log + c δ α ) ≤ λ + + 2 , and (37) Obtaining a lower estimate for A k p . First we bound A n x − A n p .
Hence we can estimate the norm as follows Since n satisfies (35) with a n (x) = log A n x , A n−(i+1) xi+1 ≤ A n x e −(i+1)(λ+− ) for all i with L ≤ i ≤ n − 1.
Using (36) and Hölder continuity of A we obtain We claim that the exponent satisfies αγ min{i, k − i} ≥ 4 i for i = 0, . . . , n.