Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles

TL;DR

This paper investigates the entropy properties of Lyapunov maximizing measures for typical $SL(2,R)$ cocycles, establishing conditions under which these measures have zero entropy, thus advancing understanding in ergodic optimization.

Contribution

It proves that Lyapunov maximizing measures for typical $SL(2,R)$ cocycles have zero entropy under specific conditions, extending previous results in ergodic optimization.

Findings

Lyapunov maximizing measures have zero entropy under certain conditions.

The maps $e_1$ and $e_2$ are one-to-one on the Mather set.

Results apply to cocycles satisfying pinching and twisting conditions.

Abstract

In this paper we study ergodic optimization problems for typical cocycles. We consider one-step $SL(2,\mathbb{R})$-cocycles that satisfy pinching and twisting conditions. We prove that the Lyapunov maximizing measures have zero entropy under additional assumptions that the maps $e_1$ and $e_2$ are one-to-one on the Mather set.