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arXiv · 1608.05757

Periodic approximation of Lyapunov exponents for Banach cocycles

Boris Kalinin·Victoria Sadovskaya
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Abstract

We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism $f$ of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We prove that upper and lower Lyapunov exponents of $A$ with respect to an ergodic invariant measure $μ$ can be approximated in terms of the norms of the values of $A$ on periodic orbits of $f$. We also show that these exponents cannot always be approximated by the exponents of $A$ with respect to measures on periodic orbits. Our arguments include a result of independent interest on construction and properties of a Lyapunov norm for infinite dimensional setting. As a corollary, we obtain estimates of the growth of the norm and of the quasiconformal distortion of the cocycle in terms of the growth at the periodic points of $f$.

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Boris Kalinin, Victoria Sadovskaya. 2016-08-19. Periodic approximation of Lyapunov exponents for Banach cocycles. https://doi.org/10.1017/etds.2017.43

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