CONTINUITY OF SPECTRAL RADIUS OVER HYPERBOLIC SYSTEMS

. The continuity of joint and generalized spectral radius is proved for H¨older continuous cocycles over hyperbolic systems. We also prove the pe- riodic approximation of Lyapunov exponents for non-invertible non-uniformly hyperbolic systems, and establish the Berger-Wang formula for general dynam- ical systems.


1.
Introduction. Let A be a compact subset of the space M d (R) of d × d real matrices. The joint spectral radius of A was defined by Rota and Strang [23] aŝ ρ(A) = lim n→∞ sup{ A n · · · A 1 1/n : A i ∈ A}. (1.1) The joint spectral radius has applications in many areas including coding theory [18] and the theory of control and stability [2,8]. Among research on the joint spectral radius, many important properties are revealed: it was proved by Wirth [27] that the function A →ρ(A) is continuous on the space of compact sets of M d (R), and is locally Lipschitz continuous on the space of irreducible compact sets of M d (R), where the explicit Lipschitz constant was given by Kozyakin [16]; the joint spectral radius can be also related to the generalized spectral radius defined by ρ(A) = lim sup n→∞ sup{ρ(A n · · · A 1 ) 1/n : where ρ(A) = lim n→∞ 1.2. Continuity of generalized spectral radius. Our next result states that the generalized spectral radius also has the continuity property. Theorem B. Let f be a continuous map of a compact metric space X, and satisfy the closing property on X. Then the function ρ s : C α (X, M d (R)) → R is continuous for every s > 0.
To deduce Theorem B, we generalize the Berger-Wang formula [4] to dynamical setting, i.e. the generalized spectral radius equals the joint spectral radius. We note that the classical Berger-Wang formula concerning the cocycle driven by full-shift system, which has also been established for finite-type sub-shift system by Dai [9]. Here we consider general dynamical systems.
Theorem C. Let f be a continuous map of a compact metric space X, A : X → M d (R) be continuous. Thenρ s (A) = ρ s (A) for every s > 0.

Lyapunov exponents and Lyapunov norm.
2.1. Lyapunov exponents. We start by studying the property of Lyapunov exponents which reflects the limit behavior of product of matrices in the dynamical processes.
Considering f is a homeomorphism of a compact metric space X preserving an ergodic measure µ, and A : X → M d (R) is continuous, the multiplicative ergodic theorem [21,12,11,1] states that there exists an f -invariant set R with µ(R) = 1, such that for each x ∈ R: (i) there exist numbers χ 1 > · · · > χ k ≥ −∞, and a measurable direct sum The numbers χ 1 > · · · > χ k are called the Lyapunov exponents of A, R d = E 1 (x)⊕· · ·⊕E k (x) is called the Oseledets decomposition, and R is called the regular set of µ.
We remark that the part n → −∞ in (iii) is proved by [ 2.2. Lyapunov norm. We assume in this subsection that k > 1, which implies . Then for a fixed ε > 0, and a point x ∈ R, we define the Lyapunov norm · x = · x,ε in R d as follows.
The next proposition gives some useful properties of the Lyapunov norm.
Proposition 2.1. Let f, A and µ be as above. Then for any fixed ε > 0, the Lyapunov norm · x = · x,ε defined above satisfies the following properties.
There exists an f -invariant set R ε ⊂ R with µ(R ε ) = 1 and a measurable function K ε (x) such that for any x ∈ R ε , Proof. (i) We will prove the inequality A(x)u E f x ≤ e χ1+ε u E x , ∀u E ∈ E(x), the others can be proved analogously.
For any x ∈ R and u E ∈ E(x), by the definition we obtain that and the inequality follows.
(ii) By the definition, u = u E +u F ≤ u E + u F ≤ u E x + u F x = u x , this gives the lower bound, so it remains to estimate the upper bound.
For any ε > 0 and x ∈ R, we define : n ≥ 0 and u F ∈ F (x) .

CONTINUITY OF SPECTRAL RADIUS OVER HYPERBOLIC SYSTEMS 3981
Then for any u = u E + u F ∈ R d with u E ∈ E(x) and u F ∈ F (x), we have Similarly, we can also obtain u F ≤ 1 sin γ(x) u . Hence we conclude by using (2.7) and (2.8) that Proof. It's enough to prove that M ε (x) and M ε (x) are tempered for µ-a.e. x ∈ R. We will prove M ε (x) is tempered for µ-a.e. x ∈ R, the other one can be proved analogously. Recall that Then the claim is proved by using [17, Lemma III.8].
Since sin γ(x) is also tempered on R by (v), we conclude by using [3, Lemma 3.5.7] that is a measurable function defined on R ε and satisfies (2.5) and (2.6). This completes the proof of Proposition 2.1.
We define then µ(R A ε,l ) → 1 as l → ∞. Moreover, without loss of generality, we may also assume by using Lusin's theorem that the Lyapunov norm and Oseledets decomposition are continuous on R A ε,l by restricting to a compact subset of it.

Approximation of Lyapunov exponents.
To prove Theorem A, we need to approximate the Lyapunov exponents of ergodic measures by those at periodic points.
Theorem 3.1. Let f be a continuous map of a compact metric space X, satisfying the closing property on a Borel subset Λ ⊂ X, µ be an ergodic invariant measure for f with µ(Λ) > 0, and A : X → M d (R) be Hölder continuous. Then the Lyapunov exponents of A with respect to µ can be approximated by the Lyapunov exponents of A at periodic points.
We note that the approximation of Lyapunov exponents of hyperbolic measures for a diffeomorphism was proved by Wang and Sun [26]. Kalinin [13] proved the approximation when f is a homeomorphism satisfying the closing property on X and A takes values in GL d (R) and furthermore, with Sadovskaya [14] considered invertible operators on a Banach space. Dai proved the approximation in [6] and [7] when f is a continuous map and A takes values in GL d (R). Backes [1] proved the approximation for semi-invertible cocycles: f is a homeomorphism and A takes values in M d (R). Theorem 3.1 deals with completely non-invertible cocycles since both f and A may be non-invertible.
To prove Theorem 3.1, we will need the following two lemmas.
. Suppose that f is a continuous map of a compact metric space X, preserves an ergodic measure µ and satisfies the closing property on a Borel subset Λ ⊂ X with µ(Λ) > 0. Then there exist periodic points Proof of Theorem 3.1. Let χ 1 > · · · > χ k ≥ −∞ be the Lyapunov exponents of A with respect to µ [21]. We will divide the proof into two cases: Let µ pj be as in Lemma 3.2, and λ 1 (A, µ pj ) ≥ · · · ≥ λ d (A, µ pj ) be the Lyapunov exponents of A with respect to µ pj counted with multiplicities. Since the function f n (x) := 1 n log A(x, n) is continuous for every n ≥ 1, and Thus we conclude that lim be the space of pre-orbits, endowed with the metriĉ

CONTINUITY OF SPECTRAL RADIUS OVER HYPERBOLIC SYSTEMS 3983
The natural extensionf :X →X of f is defined bŷ Thenf is a homeomorphism. Let π :X → X be the o th projection map, andμ be the unique ergodicf -invariant measure onX with π * μ = µ [24, Proposition it is clear that A andÂ have the same Lyapunov exponents which we denote by Sincef is invertible, by the multiplicative ergodic theorem stated in the subsec- , then the conclusion in Lemma 3.4 below also holds, and the proof is simpler. Thus we may assume k > 1). For any ε > 0, the Lyapunov norm · x = · x,ε is defined as in Denotex i =f ix ,p i =f ip , and let x = π(x), p = π(p). Let RÂ ε,l ⊂X be defined as in (2.9). For any , then by (2.5), (2.6) and if δ is small enough. Similarly, by (2.4), (3.2) and u ∈ K θ i , we can also obtain for δ small enough. Let η = e b−χ1+4ε < 1, then we have

Similar to (3.3), by (2.3) and(3.1), we can also get
for δ small enough. Thus we conclude by using (3.4) that First we will give the approximation of the largest Lyapunov exponent of A with respect to µ. Recall that µ p := 1 n n−1 i=0 δ f i p represents the periodic measure, and χ 1 (A, µ p ) = χ 1 (Â,μp) is the largest Lyapunov exponent of A with respect to µ p . Proposition 3.5. Under the assumption of "Case II", for any 0 < ε < ε 0 , there exists a periodic point p such that |χ 1 (A, µ p ) − χ 1 | ≤ 4ε. Moreover, µ p → µ in the weak * topology as ε → 0.

Proof of Theorem A.
To begin the proof of Theorem A, we first state the following lemma.
Lemma 4.1. Let f : X → X be a measure-preserving transformation of a probability space (X, B, µ), A : X → M d (R) satisfy log + A(x) ∈ L 1 (X, µ) and have Lyapunov exponents λ 1 ≥ · · · ≥ λ d with respect to µ counted with multiplicities. Then for any 0 < s < d, we have where m = s , and λ 1 (Λ i A, µ) is the largest Lyapunov exponent of Λ i A with respect to µ.
Proof. For a matrix M ∈ M d (R), by [3, Section 3.1], we have We now prove Theorem A. We assume s < d, the case s ≥ d can be considered analogously.
Since  Take µ ∈ M max (A). We may suppose that µ is ergodic, otherwise we replace µ by its ergodic component in M max (A). By Lemma 4.1, we have then B(x) is Hölder continuous. By Theorem 3.1, we deduce that the Lyapunov exponents of Λ m+1 A(x) and Λ m A(x) with respect to µ can be approximated by those at the same periodic points. Thus, Using (4.3), ) is continuous, the function A →ρ s (A) is lower semi-continuous. This completes the proof of Theorem A.

5.
Proof of Theorem C. In this section, we assume s < d . In fact, if s ≥ d, then by the definition of ρ s and ϕ s , for any matrix B ∈ M d (R), we have ρ s (B) = ϕ s (B), thus the conclusion of Theorem C is trivial.
The proof of Theorem C relies on our next result which generalizes Theorem 1.6 in [19]. Assuming Theorem 5.1 for the time being, we may give the proof of theorem C first by using Theorem 5.1.
Proof of Theorem C. For a matrix M ∈ M d (R), by (4.1), one has  This completes the proof of Theorem C.
Next we prove Theorem 5.1, which is a particular case of the following proposition.
Therefore, the conclusion of Theorem 5.1 is obtained by Lemma 4.1.