Removal of Phase Transition in the Chebyshev Quadratic and Thermodynamics for Hénon-Like Maps Near the First Bifurcation

It is well-known that the geometric pressure function (Formula presented.)of the Chebyshev quadratic map (Formula presented.)(Formula presented.) is not differentiable at (Formula presented.). We show that this phase transition can be “removed”, by an arbitrarily small singular perturbation of the map (Formula presented.) 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.