On the periodic approximation of Lyapunov exponents for semi-invertible cocycles

We prove that, for semi-invertible linear cocycles, Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.


Introduction
A very general and also vague idea that appears in the study of dynamical systems is that "if a system exhibits enough hyperbolicity then most of its dynamical interesting information is concentrated in its periodic orbits". There are many examples supporting this idea. For instance, it is known that cohomology classes of Hölder cocycles over hyperbolic systems are characterized by its information on periodic points (see for instance [14,15,10,7,1,17,2,13] and references therein), equilibrium states associated to different potentials coincide whenever those potentials have the same information on periodic points [4] and so on.
In the present work we also present an example supporting the previous "belief" in the context of Lyapunov exponents of semi-invertible linear cocycles. More precisely, we show that (see Section 2 for precise definitions and statements) Theorem 1.1. Lyapunov exponents of ergodic measures may be approximated by Lyapunov exponents on periodic points.
In other words, all the information carried by the Lyapunov exponents of (f, A) is indeed concentrated on periodic points.
The objects involved in our example are very classical in the fields of Dynamical Systems and Ergodic Theory and can be defined as follows: given an invertible ergodic measure preserving dynamical system f : M → M defined on a measure space (M, A, µ) and a measurable matrix-valued map A : M → M (d, R), the pair (f, A) is called a semi-invertible linear cocycle (or just linear cocycle for short). Sometimes one calls linear cocycle (over f generated by A), instead, the sequence {A n } n∈N defined by for all x ∈ M . The word 'semi-invertible' refers to the fact that the action of the underlying dynamical system f is invertible while the action on the fibers given by A may fail to be invertible. We refer to the Introduction of [8] for some interesting applications of semi-invertible cocycles. Under certain integrability conditions, it was proved in [9] that there exists a full µ-measure set R µ ⊂ M , whose points are called µ-regular points, such that for every x ∈ R µ there exist numbers λ 1 > . . . > λ l ≥ −∞, called Lyapunov exponents, and a direct sum decomposition R d = E 1,A x ⊕ . . . ⊕ E l,A x into vector subspaces which are called Oseledets subspaces and depend measurable on x such that, for every 1 ≤ i ≤ l, f (x) with equality when λ i > −∞ and x . This result extends a famous theorem due to Oseledets [16] known as the multiplicative ergodic theorem which was originally stated in both, invertible (both f and the matrices are assumed to be invertible) and non-invertible (neither f nor the matrices are assumed to be invertible) settings (see also [18]). While in the invertible case the conclusion is similar to the conclusion above (except that all Lyapunov exponents are finite), in the non-invertible case, instead of a direct sum decomposition into invariant vector subspaces, one only get an invariant filtration (a sequence of nested subspaces) of R d .
In the invertible setting, Theorem 1.1 was already gotten by Kalinin in [10] extending a theorem of Wang and Sun [19] on the approximation of Lyapunov exponents of hyperbolic invariant measures for diffeomorphims. In fact, the proof of our main result is based on ideas from those works. The lack of invertibility of the matrices, however, brings in some additional difficulties. To deal with it, we introduce the notion of Lyapunov norm for semi-invertible cocycles and present some useful properties about these objects.
It is also worth noticing that a similar approximation result was gotten by Dai [6] in the case when just the matrices are assumed to be invertible. More recently, Kalinin and Sadovskaya [11] addressed a similar problem in the invertible setting but when the cocycle takes values in the set of invertible operators of a Banach space. In such setting, Theorem 1.1 can not be fully recovered.

Statements
Let (M, d) be a compact metric space, µ a measure defined on the Borel sets of (M, d) and f : M → M a measure preserving homeomorphism. Assume also that µ is ergodic.
We say that f satisfies the Anosov Closing property if there exist C 1 , ε 0 , θ > 0 such that if z ∈ M satisfies d(f n (z), z) < ε 0 then there exists a periodic point p ∈ M such that f n (p) = p and for every j = 0, 1, . . . , n. Notice that shifts of finite type, basic pieces of Axiom A diffeomorphisms and more generally, hyperbolic homeomorphisms are particular examples of maps satisfying the Anosov Cloisng property. See for instance, [12] p.269, Corollary 6.4.17.
Given a continuous map A : M → M (d, R) such that log + A(x) dµ(x) < ∞, let us denote by the Lyapunov exponents of the cocycle (f, A) with respect to the measure µ and by the Lyapunov exponents of (f, A) with respect to µ counted with multiplicities. Given a periodic point p, we denote its Lyapunov exponents and Lyapunov exponents counted with multiplicities by respectively. When there is no risk of ambiguity, we suppress the index A or even both A and µ from the previous objects.
In what follows we are also going to assume that A : M → M (d, R) is an α-Hölder continuous map. This means that there exists a constant C 2 > 0 such that for all x, y ∈ M where A denotes the operator norm of a matrix A, that is, As a simple consequence of our main result we get the following corollary. In order to state it, let us assume that f and A satisfy the hypotheses of Theorem 2.1. Then, Corollary 2.2. If all Lyapunov exponents of (f, A) on periodic points are uniformly bounded by bellow then A(x) ∈ GL(d, R) for almost every x ∈ M with respect to every f -invariant probability measure µ.
Proof. If there exist a measure µ, which by the the Ergodic Decomposition Theorem may be assumed to be ergodic, and a set B ⊂ M with positive µ-measure such that A(x) / ∈ GL(d, R) for every x ∈ B then γ d (A, µ) = −∞ which in light of Theorem 2.1 contradicts our assumption.
We observe that satisfying A(x) ∈ GL(d, R) for almost every x ∈ M with respect to every f -invariant probability measure as in the previous corollary does not imply, in general, that A(x) ∈ GL(d, R) for every x ∈ M . Indeed, Let q = (q i ) i∈Z ∈ M be such that q i = 1 for every i = −1 and q −1 = 0 and fix a > 1 so that aθ > 1. Consider A : M → R given by Then, A is a Hölder map and generates a semi-invertible cocycle over f . Moreover, A(q) = 0 and (f, A) has all Lyapunov exponents on periodic points uniformly bounded by bellow by log(aθ) > 0. In fact, if a periodic point p ∈ M is such that d(f j (p), q) > θ 3 for every j ∈ N then obviously λ 1 (p) = log a > 0. Now, suppose there exists j ∈ N so that d(f j (p), q) ≤ θ 3 . We may assume without loss of generality that j = 0. More precisely, suppose p = (p i ) i∈Z satisfies p i = q i for every | i |≤ n and p i = q i for some i so that | i |= n + 1 with n ≥ 2. Then, A(p) = a θ 3 θ n+1 and A(f j (p)) = a for, at least, j = 1, 2, . . . , n + 1. In particular, A n+2 (p) = a n+2 θ n−2 and 1 n+2 log A n+2 (p) ≥ log(aθ) > 0. In other words, whenever a periodic point p is θ n+1 close to q, its next n + 1 iterations are going to be out of the ball of radius θ 3 and centered at q. Repeating this argument we can construct a sequence (n k ) k∈N going to +∞ such that 1 Observe that, besides the choice of A, the main feature underlying our construction is that q ∈ W s (p)∩W u (p) where p is the fixed point p = (p i ) i∈Z such that p i = 1 for every i ∈ Z. Another simple remark is that such example can be constructed in any dimension. We take A : M → M (d, R) to be such that A(x) is a diagonal matrix for every x ∈ M where one of its entries is the map constructed above while the others are all constant.
The previous example also reveals very different behaviors of invertible and semiinvertible cocycles. Indeed, it was proved by Cao in [5] that if an invertible cocycle A defined over a homeomorphism f has only positive Lyapunov exponents with respect to every f -invariant probability measure then it is uniformly expanding. The previous example shows us that this is no longer true in the semi-invertible context.

Lyapunov norm
In order to estimate the growth of the cocycle A along an orbit we introduce the notion of Lyapunov norm for semi-invertible cocycles. This is based on a similar notion for invertible cocycles (see for instance [3]).
Let x ∈ R µ be a regular point and x be the Oseledets decomposition at x. Given i ∈ {1, . . . , l − 1} and n ∈ N, let us consider the map which is invertible and let us denote its inverse by (A n (f −n (x))) −1 i . Now, for every n ∈ Z and x ∈ R µ let us consider the linear map Observe that, for every m, n ∈ Z, Indeed, for m, n ≥ 0 it follows readily from the definition and (1). Suppose now m, n > 0 and let us prove that . We start observing that, since , it follows by the invariance of the Oseledets spaces that Thus, taking the inverses on both sides the result follows. The case when m and n have different signs may be deduced by combining the previous two.
In order to define the Lyapunov norm associated to the cocycle A at a regular point x ∈ R µ , we start by defining the Lyapunov inner product : given δ > 0 and two vectors u = u 1 + . . .
for every i for which λ i is finite and in the case when λ l = −∞. Observe that both series (3) and (4) converge for any x ∈ R µ . Indeed, convergence of the second one is easily to verify while the convergence of the first one follows from the next lemma whose proof is also going to be used in the sequel.
Proof. The fact that lim n→+∞ 1 n log A n i (x)u = λ i follows from the definition and Oseledets' Theorem. Let us prove that lim n→−∞ for any n ∈ Z. Combining this two inequalities we get that 1 Taking the logarithm at each term, dividing by −n and making n → +∞ it follows that . Now, since ε > 0 is arbitrary the lemma follows.
In particular, it follows that ·, · x,δ is actually an inner product in R d . We then define the δ-Lyapunov norm . x,δ associated to the cocycle A at x ∈ R µ as the norm generated by ·, · x,δ . When there is no risk of ambiguity, we just write . x and . x,i instead of . x,δ and . x,δ,i and call it just Lyapunov norm.
Given a liner map B ∈ M (d, R), its Lyapunov norm is defined for any regular points x, y ∈ R µ by The next proposition gives us some useful properties of the Lyapunov norm that we are going to use in the sequel. In the case when the cocycle is invertible, similar properties of the Lyapunov norm are very well known. We now prove them in the semi-invertible setting.
for every n ∈ N; ii) If λ l = −∞ then for every u ∈ E l,A x and n ∈ N we have for every n ∈ N; iv) For every δ > 0 such that − 1 δ < λ l−1 , there exists a measurable function whose growth along any regular orbit is bounded; more precisely, Consequently, for any linear map B and any regular points x and y Proof. In order to prove i) we observe that for any u ∈ E i,A x , Consequently, for every u ∈ E l,A x . In order to get iii) one only have to observe that, for any u ∈ R d , The first inequality in iv) is trivial. To prove the second one, we proceed analogously to what we did in Lemma 3.1. Fix i ∈ {1, . . . , l − 1} and ε > 0 such that 2ε < δ. From Theorem 2 of [8] it follows that there exists a measurable map C : M → (0, ∞) such that 1 C(x) e (λi−ε)n u ≤ A n i (x)u ≤ C(x)e (λi +ε)n u for every n ∈ N and u ∈ E i,A x . Equation (5) from the proof of Lemma 3.1 tells us that 1 C(x) e −(λi+2ε)n u ≤ A −n i (x)u ≤ C(x)e −(λi−2ε)n u for every n ∈ N. Combining this two equations we get that For u ∈ E l,A x , Theorem 2 of [8] tells us that whenever − 1 δ < λ l−1 , Thus, Thus, taking K = max{ n∈Z e (4ε−2δ)|n| , n≥0 e − 4 δ n } and writing u ∈ R d as It remains to obtain an upper bound for u i in terms of u . This can be achieved by using the map K given by Theorem 2 of [8]. More precisely, let K 1 be the map given by [8, Theorem 2] applied for i = 1 and sufficiently small ǫ > 0. We then have that The first inequality in (11) gives a desired bound for u 1 . In order to obtain the bound for u 2 , we can apply again [8, Theorem 2] but now for i = 2 (and again for ǫ > 0 sufficiently small) to conclude that there exists K 2 such that By combining the second inequality in (11) with the first inequality in (12), we conclude that u 2 ≤ K 1 (x)K 2 (x) u . By proceeding, one can establish desired bounds for all u j , j = 1, . . . , l and construct a function K δ satisfying (9). Indeed, this follows from the fact that C(f n (x)) ≤ C(x)e ε|n| for every n ∈ Z and similarly for the maps K 1 , K 2 , . . . , K l completing the proof.
For any N > 0, let R µ δ,N be the set of regular points x ∈ R µ for which K δ (x) ≤ N . Observe that µ(R µ δ,N ) → 1 as N → +∞. Moreover, invoking Lusin's theorem we may assume without loss of generality that this set is compact and that the Lyapunov norm and the Oseledets splitting are continuous when restricted to it.
As a final and simple remark about Lyapunov norms we observe that, in order to get a norm satisfying the properties given by the previous proposition, it is not necessary to use inner products. Indeed, for x ∈ R µ , δ > 0 and u = u 1 +. Approximation of the largest Lyapunov exponent. We start the proof of Theorem 2.1 with a key proposition which tells us that the largest Lyapunov exponent of A may be approximated by Lyapunov exponents on periodic points. We retain all the notation introduced at the previous section.
More precisely, there exists δ 0 > 0 such that for any N > 0 and δ ∈ (0, δ 0 ), there exist n 0 ∈ N and ρ > 0 such that, for every n ≥ n 0 , if x, f n (x) ∈ R µ δ,N are such that d(x, f n (x)) < ρ and p is a periodic point associated to x by the Anosov Closing property then | λ 1 − λ 1 (p) |< δ.
Let C 1 , ε 0 , θ > 0 be given by the Anosov Closing Property, ρ ∈ (0, ε 0 ) and suppose d(x, f n (x)) < ρ. Thus, there exists a periodic point p ∈ M of period n such that for every j = 0, 1, . . . , n. We will prove that, as long as n is sufficiently large, this periodic point satisfies the previous proposition. We split the proof into two lemmas. In the first one we give a lower bound for λ 1 (p) in terms of λ 1 while in the second one an upper bound is given.

Lemma 4.2 (Lower bound).
There exists n 0 ∈ N such that, if ρ ∈ (0, ε 0 ) is sufficiently small and n ≥ n 0 then Proof. For each 1 ≤ j ≤ n, let us consider the splitting and u j F ∈ F f j (x) . Then the cone of radius 1 − γ > 0 around E 1,A f j (x) is defined as .
To simplify notation we write . j for the Lyapunov norm at the point f j (x).
We claim now that it is enough to prove that if ρ is sufficiently small then there exists γ ∈ (0, 1) such that and for every u ∈ C j 0 , Indeed, since we are assuming that the Oseledets splitting and the Lyapunov norm are continuous on R µ δ,N , it follows that if ρ is sufficiently small (and consequently x and f n (x) are close) then C n γ ⊂ C 0 0 and thus by (15), A n (p)(C 0 0 ) ⊂ C 0 0 . Consequently, for any u ∈ C 0 0 and k ∈ N we have that A kn (p)u ∈ C 0 0 . Therefore, given u ∈ C 0 0 , invoking (16) and the fact that the Lyapunov norms at x and f n (x) are close whenever ρ is small, which applied k times leads us to Consequently, as long as n is large enough which proves our claim. So, the only thing left to do is to prove (15) and (16). Assume initially that λ 2 > −∞.
Given u ∈ C j 0 let us consider v = A(f j (x))u. Then, it follows from (6) that v j+1 ≤ e λ1+δ u j and moreover that v j+1 Let w = A(f j (p))u. What we want to do now is to compare the Lyapunov norms of w and its projection on E 1,A f j+1 (x) and F f j+1 (x) with the respective norms of v. In order to do it, let us consider B j = A(f j (p)) − A(f j (x)). Consequently, w = v + B j u and thus w j+1 F . Moreover, for every 0 ≤ j ≤ n, Therefore, invoking (10) and (8) it follows that Then, using that K δ (f j+1 (x)) ≤ N e δ min{j+1,n−j−1} which follows from (9) and the fact that x and f n (x) are in R µ δ,N and that u j ≤ 2 u j − B j u j+1 Therefore, combining this inequalities and using again that Taking ρ small enough so that e λ1−δ − e λ2+δ − 2Cρ α > 0 and applying (18) to the previous inequality we get that there exists γ > 0 such that w j+1 which implies that w = A(f j (p))u ∈ C j+1 γ proving (15) and consequently the lemma whenever λ 2 > −∞. The case when λ 2 = −∞ is analogous. The only difference is that inequality (17) becomes
Consequently, if ρ > 0 is sufficiently small and n is large enough, Proof. Let us consider B j = A(f j (p)) − A(f j (x)). As in the proof of the previous lemma we have that, for every 0 ≤ j ≤ n, Our objective now is to estimate A n (p) f n (x)←x . We start observing that and, for each 0 ≤ j < n, Thus, since from (7) invoking (21) we get that Now, using the fact that δ − θα < 0 and making c = exp C ∞ j=0 2e (δ−θα)j we get the first claim of the lemma. The second one follows from the previous one observing that x ∈ R µ δ,N and A n (p) ≤ K δ (x) A n (p) f n (x)←x . To conclude the proof it only remains to observe that which combined with the previous inequality implies λ 1 (p) ≤ λ 1 + δ + 1 n log(cN e ρ α ).
Since ρ may be taken arbitrary small the lemma follows.
Proposition 4.1 now follows easily from these two lemmas.

4.2.
Approximation of the infinite Lyapunov exponent. In this subsection we prove that even when the Lyapunov exponents are not finite we still can approximate it by Lyapunov exponents on periodic points. This will follow from the next general proposition. We start observing that, since µ is ergodic, ϕ + 1 (x) = max{0, ϕ 1 (x)} ∈ L 1 (µ) and {nϕ n } n∈N is a subadditive sequence, it follows by Kingman's Subadditive Theorem (see for instance [18]) that for µ almost every x ∈ M . Analogously, since for each j ∈ N the measure µ j is ergodic and ϕ + 1 (x) ∈ L 1 (µ j ), we have Observing then that, as in (22) We may assume without loss of generality that µ is non-atomic. Otherwise the theorem is trivial. Moreover, we assume λ l = −∞. In the case when λ l > −∞ we only need the first part of our argument. Recall that γ 1 ≥ γ 2 ≥ . . . ≥ γ d are the Lyapunov exponents of A with respect to µ counted with multiplicities and, for every i ∈ {1, . . . , d}, let Λ i (R d ) be the ith exterior power of R d which is the space of alternate i-linear forms on the dual (R d ) * and Λ i A(x) : Λ i (R d ) → Λ i (R d ) the cocycle induced by A(x) on the ith exterior power. A very well known fact about this cocycle (see for instance [18]) is that its Lyapunov exponents are {γ j1 + . . . + γ ji ; 1 ≤ j 1 < . . . < j i ≤ d}.
Let N be large enough so that the intersection G of the sets R µ δ,N associated to all the cocycles Λ i A for i = 1, . . . , dim(E 1,A By the Anosov Closing property it follows that, for each k sufficiently large, there exists a periodic point p k of period n k so that d(f j (x), f j (p k )) ≤ C 1 e −θ min{j,n k −j} d(f n k (x), x) ≤ C 1 k e −θ min{j,n k −j} (23) for every j = 0, 1, . . . , n k . It follows then by Proposition 4.1 applied to the cocycles Λ i A for i = 1, . . . , dim(E 1,A x ⊕ . . . ⊕ E l−1,A x ) that for every δ > 0 small, there exists k δ ∈ N so that for any k ≥ k δ , | (γ 1 + γ 2 + . . . + γ i ) − (γ 1 (p k ) + γ 2 (p k ) + . . . + γ i (p k )) |< δ ) and that the sequence (µ p k ) k∈N of ergodic periodic measures given by converges to µ in the weak * topology which follows from the fact that x ∈ B(µ) and (23). Consequently, lim k→+∞ γ i (p k ) = γ i for every i = 1, . . . , d completing the proof of Theorem 2.1.