Equilibrium Measures at Temperature Zero for Hénon-Like Maps at the First Bifurcation

Abstract

We develop a thermodynamic formalism for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$ we prove the existence of an invariant Borel probability measure which minimizes the free energy associated with a noncontinuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. We characterize accumulation points of these measures as $t\to\pm\infty$ in terms of the unstable Lyapunov exponent.