Removal of phase transition of the Chebyshev quadratic and thermodynamics of H\'enon-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 $$t\in\mathbb R\mapsto \sup_{\mu}\left\{h_\mu(T_2)-t\int\log |dT_2(x)|d\mu(x)\right\}$$ 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.