Lyapunov optimizing measures and periodic measures for $C^2$ expanding maps

TL;DR

This paper proves that for a generic set of smooth expanding maps on the circle, the measures minimizing Lyapunov exponents are uniquely supported on periodic orbits, confirming a conjecture in the $C^2$ topology.

Contribution

It establishes that in the space of $C^2$ expanding circle maps, Lyapunov minimizing measures are generically supported on periodic orbits, resolving a conjecture by Jenkinson-Morris.

Findings

Lyapunov minimizing measures are generically supported on periodic orbits.

The result holds for an open and dense subset in the $C^2$ topology.

Confirms the Jenkinson-Morris conjecture in the $C^2$ setting.

Abstract

We prove that there exists an open and dense subset $\mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.