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 fibres corresponding to the central direction. The sets are genuinely non-hyperbolic, containing intermingled horseshoes of different hyperbolic, behaviour (contracting and expanding centre). We construct for every k ⩾ 1 a diffeomorphism F with a transitive set Λ as above such that the pressure map P(t) = P(t φ) of the potential $\varphi= -\log \,\Vert \rmd F|_{E^c}\Vert$ (E^c the central direction) defined on Λ has k rich phase transitions. This means that there are parameters t_ℓ, ℓ = 0, ..., k − 1, where P(t) is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of t_ℓ φ with positive entropy and different Birkhoff averages. Each phase transition is associated with a gap in the central Lyapunov spectrum of F on Λ.