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 t ∈ R ↦ sup μ h μ ( T 2 ) - t ∫ log | d T 2 ( x ) | d μ ( x ) of the Chebyshev quadratic map T 2 ( x ) = 1 - 2 x 2 ( x ∈ 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.