Paper detail

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 $k\ge 1$ 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 $φ= -\log \,\lVert dF|_{E^c}\rVert$ ($E^c$ the central direction) defined on $Λ$ has $k$ rich phase transitions. This means that there are parameters $t_\ell$, $\ell=1,...,k$, where $P(t)$ is not differentiable and this lack of differentiability is due to the coexistence of two equilibrium states of $t_\ell\,φ$ with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum of $F$ on $Λ$.