Ergodic optimization theory for Axiom A flows

TL;DR

This paper studies ergodic optimization for Axiom A flows, showing that for generic observables, the minimizing measure is unique and supported on a periodic orbit, advancing understanding of dynamical systems.

Contribution

It establishes the generic uniqueness and periodic support of minimizing measures for weighted ergodic optimization in Axiom A flows.

Findings

For generic observables, the minimizing measure is unique.

The minimizing measure is supported on a periodic orbit.

Results apply to functions in $ ext{C}^{0,eta}$ and $ ext{C}^1$ spaces.

Abstract

In this article, we consider the weighted ergodic optimization problem Axiom A attractors of a $C^2$ flow on a compact smooth manifold. The main result obtained in this paper is that for a generic observable from function space $\mc C^{0,\a}$ ($\a\in(0,1]$) or $\mc C^1$ the minimizing measure is unique and is supported on a periodic orbit.