Maximizing measures for expanding transformations

TL;DR

This paper investigates maximizing measures for expanding transformations, proving generic uniqueness for Holder functions, approximation by periodic orbit measures, and convergence of equilibrium states, with applications to circle endomorphisms.

Contribution

It establishes generic uniqueness of maximizing measures for Holder functions and shows how these measures can be approximated by periodic orbit measures and equilibrium states.

Findings

Generic Holder functions have unique maximizing measures.

Maximizing measures can be approximated by periodic orbit measures.

Lyapunov maximizing measures are generically supported on periodic orbits.

Abstract

On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function whose maximizing measure is supported on a periodic orbit. We also show that maximizing measures can be obtained as weak limits of equilibrium states. We apply these theorems to the class of C(1+a) endomorphisms of the circle which are coverings of degree 2, uniformly expanding and orientation preserving. We prove that generically the invariant probability which maximizes the Lyapunov exponent is unique and that C(1+b) endomorphisms can be approximated in the C(1+a) topology by endomorphisms whose Lyapunov maximizing measure is supported on a periodic orbit.