TY - JOUR

T1 - Periodic approximation of Lyapunov exponents for Banach cocycles

AU - Kalinin, Boris

AU - Sadovskaya, Victoria

N1 - Publisher Copyright:
© 2017 Cambridge University Press.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism 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 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 the 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.

AB - We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism 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 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 the 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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U2 - 10.1017/etds.2017.43

DO - 10.1017/etds.2017.43

M3 - Article

AN - SCOPUS:85021094038

VL - 39

SP - 689

EP - 706

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -