Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties

Abstract

For strongly dissipative Hénon 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.

39 pages, 8 figures. Ergodic Theory and Dynamical Systems, to appear

References in corpus (2)

• The boundary of hyperbolicity for Henon-like families
• Equilibrium measures for the Hénon map at the first bifurcation

Cited by in corpus (3)

• Thermodynamics of Towers of Hyperbolic Type
• Thermodynamics of the Katok Map
• Removal of phase transition of the Chebyshev quadratic and thermodynamics of Hénon-like maps near the first bifurcation