Equilibrium measures for the H\'enon map at the first bifurcation: uniqueness and geometric/statistical properties

TL;DR

This paper studies the Hénon map at the bifurcation point, proving the existence and uniqueness of equilibrium measures with specific geometric and statistical properties, despite the loss of hyperbolicity.

Contribution

It establishes a thermodynamic formalism for the Hénon map at bifurcation, including existence and uniqueness of equilibrium measures for a non-continuous potential.

Findings

Existence of a unique equilibrium measure for the geometric potential

Characterization of geometric properties of the measures

Analysis of statistical properties of the measures

Abstract

For strongly dissipative H\'enon maps at the first bifurcation where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e., prove the existence and uniqueness of an invariant probability measure which maximizes the free energy associated with a non continuous geometric potential $-t\log J^u$, where $t\in\mathbb R$ is in a certain large interval and $J^u$ is the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.