# Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows

## TL;DR

This paper proves that equilibrium states for rank 1 geodesic flows are mixing and Bernoulli, establishing their strong statistical properties and developing new techniques for uniqueness in product systems with center directions.

## Contribution

It demonstrates the Kolmogorov and Bernoulli properties of equilibrium states for rank 1 geodesic flows and introduces methods for uniqueness in systems with 2D center directions.

## Key findings

- Equilibrium states are mixing of all orders.
- The Bowen-Margulis measure is Bernoulli.
- New techniques for uniqueness in product flows with center directions.

## Abstract

Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently by Burns, Climenhaga, Fisher and Thompson. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure then the equilibrium state is unique. The main result of this paper is that these equilibrium states have the Kolmogorov property. In particular, these measures are mixing of all orders and have positive entropy. For the Bowen-Margulis measure, we go further and obtain the Bernoulli property from the K-property using classic arguments from Ornstein theory. Our argument for the K-property is based on an idea due to Ledrappier. We prove uniqueness of equilibrium states on the product of the system with itself. To carry this out, we develop techniques for uniqueness of equilibrium states which apply in the presence of the 2-dimensional center direction which appears for a product of flows. This is a key technical challenge of this paper.