Equilibrium measures at temperature zero for H\'enon-like maps at the first bifurcation

TL;DR

This paper develops a thermodynamic formalism for a dissipative Hénon-like map at the first bifurcation, analyzing equilibrium measures and their convergence properties related to Lyapunov exponents.

Contribution

It introduces a new thermodynamic framework for Hénon-like maps at bifurcation, characterizing equilibrium measures and their limits as parameters vary.

Findings

Existence of invariant measures minimizing free energy for all real t.

Measures as t→+∞ converge to those minimizing unstable Lyapunov exponent.

Measures as t→−∞ converge to a Dirac measure maximizing the unstable Lyapunov exponent.

Abstract

We develop a thermodynamic formalism for a strongly dissipative H\'enon-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.