Ground States are generically a Periodic Orbit

TL;DR

This paper proves that for generic Lipschitz functions on expanding transformations, the measures that maximize the integral are supported on a single periodic orbit, highlighting a typical dynamical behavior.

Contribution

It establishes that generically, maximizing measures for Lipschitz functions are supported on periodic orbits in expanding systems.

Findings

Maximizing measures are supported on a single periodic orbit.

This behavior is generic among Lipschitz functions.

Supports the understanding of ergodic optimization in expanding dynamics.

Abstract

We prove that for an expanding transformation the maximizing measures of a generic Lipschitz function are supported on a single periodic orbit.