Low complexity of optimizing measures over an expanding circle map

TL;DR

This paper proves that for real analytic expanding circle maps, optimizing measures typically have zero entropy unless the potential is cohomologous to a constant, using symbolic space group structure to address transversality.

Contribution

It establishes a zero entropy property for optimizing measures in real analytic expanding circle maps, revealing a new structural insight and solving a transversality problem.

Findings

Optimizing measures have zero entropy unless the potential is cohomologous to constant.

The group structure of symbolic space is used to solve a transversality problem.

Applications to Lyapunov optimizing measures and generic smooth potentials are discussed.

Abstract

In this paper, we prove that for real analytic expanding circle maps, all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to constant. We use the group structure of the symbolic space to solve a transversality problem involved. We also discuss applications to optimizing measures for generic smooth potentials and to Lyapunov optimizing measures.