Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Abstract

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.

A dynamical Thouless formula

“…locally constant cocycles over a Bernoulli or Markov shift) are Hölder continuous assuming a generic irreducibility condition (see [28] and [14,Chapter 5]) and weak-Hölder continuous without such an assumption (see [17]). Under appropriate conditions, Lyapunov exponents of linear cocycles over uniformly hyperbolic systems are Hölder continuous (see [18]). Similar continuity properties were also obtained for mixed random-quasiperiodic cocycles (see [10]).…”

Construction and applications of proximal maps for typical cocycles

“…Typicality has first been introduced by Bonatti and Viana [BV04] to establish the simplicity of the Lyapunov exponents for any invariant measure with continuous local product structure. Later, it was shown that typical cocycles possess other interesting properties, including the uniqueness of the equilibrium states for the singular value potentials [Par20] and the validity of various limit laws [PP20,DKP21]. The following theorem is another interesting property that can be derived from the typicality assumption.…”