Abundant Rich Phase Transitions in Step Skew Products

We study phase transitions for the topological pressure of geometric potentials of transitive sets. The sets considered are partially hyperbolic having a step skew product dynamics over a horseshoe with one-dimensional fibers corresponding to the central direction. The sets are genuinely non-hyperbolic containing intermingled horseshoes of different hyperbolic behavior (contracting and expanding center). We prove that for every k ≥ 1 there is a diffeomorphism F with a transitive set Λ as above such that the pressure map P (t) = P (t φ) of the potential φ = − log ‖dF |Ec‖ (E the central direction) defined on Λ has k rich phase transitions. This means that there are parameters t`, ` = 1, . . . , k, where P (t) is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of t` φ with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum of F on Λ.