# Uniqueness of the measure of maximal entropy for singular hyperbolic flows in dimension 3 and more results on equilibrium states

## TL;DR

This paper proves the uniqueness of the measure of maximal entropy and equilibrium states for 3D singular hyperbolic attractors, establishing conditions under which these measures are regular and unique.

## Contribution

It establishes the uniqueness of equilibrium states for singular hyperbolic attractors in three dimensions and provides conditions to exclude singularities as equilibrium states.

## Key findings

- Unique measure of maximal entropy exists for 3D singular hyperbolic attractors.
- Conditions are given to ensure equilibrium states are regular measures.
- No singularity can be an equilibrium state under certain potential conditions.

## Abstract

We prove that any 3-dimensional singular hyperbolic attractor admits for any H\"older continuous potential $V$ at most one equilibrium state for $V$ among regular measures. We give a condition on $V$ which ensures that no singularity can be an equilibrium state. Thus, for these $V$'s, there exists a unique equilibrium state and it is a regular measure. Applying this for $V\equiv 0$, we show that any 3-dimensional singular hyperbolic attractor admits a unique measure of maximal entropy.