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

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.