Removal of phase transition of the Chebyshev quadratic and thermodynamics of H\'enon-like maps near the first bifurcation

TL;DR

This paper demonstrates that the phase transition in the geometric pressure function of the Chebyshev quadratic map can be eliminated through a small perturbation into Hénon-like maps, using advanced inducing techniques.

Contribution

It introduces a method to remove phase transitions in the pressure function by perturbing Chebyshev maps into Hénon-like diffeomorphisms near bifurcation points.

Findings

Phase transition at t=-1 can be removed with small perturbations.

Inducing techniques are adapted for Hénon-like dynamics.

Small perturbations eliminate non-differentiability in pressure function.

Abstract

We treat a problem at the interface of dynamical systems and equilibrium statistical physics. It is well-known that the geometric pressure function $$t\in\mathbb R\mapsto \sup_{\mu}\left\{h_\mu(T_2)-t\int\log |dT_2(x)|d\mu(x)\right\}$$ 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 H\'enon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to H\'enon-like dynamics near the first bifurcation.