Removal of phase transition of the Chebyshev quadratic and thermodynamics of Hénon-like maps near the first bifurcation

Abstract

We treat a problem at the interface of dynamical systems and equilibrium statistical physics. It is well-known that the geometric pressure function $$t\in \mathbb{R}\mapsto \underset{\mu }{\sup }\left\{h_{\mu }(T_2)-t\int \log |d{T}_2(x)|d\mu (x)\right\}$$ of the Chebyshev quadratic map $T_2(x)=1-2x^2$ $(x\in \mathbb{R})$ is not differentiable at $t=-1$. We show that this phase transition can be "removed", by an arbitrarily small singular perturbation of the map $T_2$ into Hénon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to Hénon-like dynamics near the first bifurcation.

24 pages, 6 figures

References in corpus (4)

• The boundary of hyperbolicity for Henon-like families
• Renormalisation-induced phase transitions for unimodal maps
• Equilibrium measures for the Hénon map at the first bifurcation
• Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties