# Ergodic optimization for hyperbolic flows and Lorenz attractors

## TL;DR

This paper investigates ergodic optimization in hyperbolic flows and Lorenz attractors, establishing generic properties of maximizing measures and linking flow and map optimization techniques.

## Contribution

It introduces new results on the uniqueness and nature of maximizing measures for hyperbolic flows and Lorenz attractors, connecting flow and Poincaré map optimization.

## Key findings

- Generic continuous observables have a unique maximizing measure with full support and zero entropy.
- Generic Holder continuous observables have a unique, periodic maximizing measure.
- Results extend to Lorenz attractors via approximation by hyperbolic sets.

## Abstract

In this article we consider the ergodic optimization for hyperbolic flows and Lorenz attractors with respect to both continuous and Holder continuous observables. In the context of hyperbolic flows we prove that a Baire generic subset of continuous observables have a unique maximizing measure, with full support and zero entropy, and that a Baire generic subset of Holder continuous observables admit a unique and periodic maximizing measure. These results rely on a relation between ergodic optimization for suspension semiflows and ergodic optimization for the Poincar\'e map with respect to induced observables, which allow us to reduce the problem for the context of maps. Using that singular-hyperbolic attractors are approximated by hyperbolic sets, we obtain related results for geometric Lorenz attractors.