On the relation between Lyapunov exponents and exponential decay of correlations

TL;DR

This paper explores the relationship between Lyapunov exponents and the exponential decay of correlations in chaotic one-dimensional maps, providing explicit bounds for certain classes of maps.

Contribution

It establishes explicit bounds linking Lyapunov exponents and decay rates for piecewise linear expanding Markov maps, extending insights to more general maps.

Findings

Explicit bounds for decay rates in terms of Lyapunov exponents for specific maps

Demonstrates the relation between chaos measures and correlation decay

Comments on generalizations to broader classes of maps

Abstract

Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates are related. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps.