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| We study Lyapunov exponents for a family of partially hyperbolic and topologically transitive local diffeomorphisms that are step skew-products over a horseshoe map. These maps are genuinely non-hyperbolic and the central Lyapunov spec trum contains negative and positive values. We show that in a first model, besides one gap, this spectrum is complete.
The principal ingredients of our proofs are minimality of the underlying iterated function system and shadowing- like arguments. In another model we study multiple phase transitions for the topological pre
ssure of geometric potentials. We prove that for every $k$ there is a diffeomorphism with a transitive set as above such that the pressure map of the parametrized geometric potential has $k$ rich phase transitions. This means that there are $k$ parameters where pressure is not differentiable and this lack
of differentiability is due to the coexistence of two equilibrium states with positive entropy and different Birkhoff averages. Each phase transition is associated to a gap in the central Lyapunov spectrum |
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