### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 689-706 |

Number of pages | 18 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics