Equilibrium measures for the H\'enon map at the first bifurcation

TL;DR

This paper investigates the existence of equilibrium measures for the strongly dissipative Hénon map at the first bifurcation point where hyperbolicity breaks down due to tangencies, focusing on a specific non-continuous potential.

Contribution

It establishes the existence of an equilibrium measure minimizing free energy for the Hénon map at a critical bifurcation, addressing a complex dynamical transition.

Findings

Existence of equilibrium measure at bifurcation point

Minimization of free energy for non-continuous potential

Characterization of dynamics near the first bifurcation

Abstract

We study the dynamics of strongly dissipative H\'enon maps, at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove the existence of an equilibrium measure which minimizes the free energy associated with the non continuous potential $-t\log J^u$, where $t\in\mathbb R$ is in a certain interval of the form $(-\infty,t_0)$, $t_0>1$ and $J^u$ denotes the Jacobian in the unstable direction.