Transfer operators and limit laws for typical cocycles

TL;DR

This paper demonstrates that typical cocycles over irreducible subshifts of finite type follow several limit laws, such as the central limit theorem and large deviations, with their Lyapunov exponents analytically depending on equilibrium states.

Contribution

It establishes limit laws for typical cocycles using transfer operators and analyzes the analytic dependence of Lyapunov exponents on equilibrium states.

Findings

Typical cocycles obey the central limit theorem.

Large deviation principles hold for these cocycles.

Lyapunov exponents depend analytically on equilibrium states.

Abstract

We show that typical cocycles (in the sense of Bonatti and Viana) over irreducible subshifts of finite type obey several limit laws with respect to the unique equilibrium states for H\"older potentials. These include the central limit theorem and the large deviation principle. We also establish the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state. The transfer operator and its spectral properties play key roles in establishing these limit laws.