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 non continuous geometric potential $-t\log J^u$, where $J^u$ denotes the Jacobian in the unstable direction. Under a mild condition, we show that any accumulation point of these measures as $t\to +\infty$ minimizes the unstable Lyapunov exponent. We also show that the equilibrium measures converge as $t\to -\infty$ to a Dirac measure which maximizes the unstable Lyapunov exponent.

16 pages, 3 figures

References in corpus (1)

• Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes