Equilibrium measures for the Hénon map at the first bifurcation

Samuel Senti, Hiroki Takahasi

Published 2011-10-04, updated 2013-03-13Version 4

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.