# Lyapunov optimization for non-generic one-dimensional expanding Markov maps

## TL;DR

This paper investigates Lyapunov optimizing measures for certain non-generic expanding Markov maps, revealing the existence of both uncountably many ergodic measures with positive entropy and unique periodic orbit measures, using novel perturbation techniques.

## Contribution

It introduces a new $C^1$ perturbation lemma enabling interpolation between expanding Markov maps and shift maps, advancing understanding of Lyapunov optimization in non-generic settings.

## Key findings

- Existence of uncountably many Lyapunov optimizing measures with positive entropy.
- Existence of unique periodic orbit optimizing measures in certain non-generic cases.
- Development of a new perturbation lemma for $C^1$ expanding Markov maps.

## Abstract

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some H\"older continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^1$ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.