Equilibrium measures for the H\'enon map at 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.