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phase transitions in step skew products
"... Abstract. We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exp ..."
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Abstract. We present examples of partially hyperbolic and topologically transitive local diffeomorphisms defined as skew products over a horseshoe which exhibit rich phase transitions for the topological pressure. This phase transition follows from a gap in the spectrum of the central Lyapunov exponents. It is associated to the coexistence of two equilibrium states with positive entropy. The diffeomorphisms mix hyperbolic behavior of different types. However, in some sense the expanding behavior is not dominating which is indicated by the existence of a measure of maximal entropy with nonpositive central exponent. 1.
Porcupinelike horseshoes: transitivity, Lyapunov spectrum, and phase transitions
 Fund. Math
"... Abstract. We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skewproduct over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The as ..."
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Abstract. We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skewproduct over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many nontrivial fibers. Moreover, the spectrum of the central Lyapunov exponents of F Λ contains a gap and hence gives rise to a first order phase transition. A major part of the proofs relies on the analysis of an associated iterated function system that is genuinely noncontracting. Contents
PORCUPINELIKE HORSESHOES. TOPOLOGICAL AND ERGODIC ASPECTS
"... Abstract. We introduce a class of topologically transitive and partially hyperbolic sets called porcupinelike horseshoes. The dynamics of these sets is a step skew product over a horseshoe. The fiber dynamics is given by a onedimensional genuinely noncontracting iterated function system. We stud ..."
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Abstract. We introduce a class of topologically transitive and partially hyperbolic sets called porcupinelike horseshoes. The dynamics of these sets is a step skew product over a horseshoe. The fiber dynamics is given by a onedimensional genuinely noncontracting iterated function system. We study this dynamics and explain how the properties of the iterated function system can be translated to topological and ergodic properties of the porcupines.
ABUNDANT RICH PHASE TRANSITIONS IN STEP SKEW PRODUCTS
"... Abstract. 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 onedimensional fibers corresponding to the central direction. The sets are genuinely n ..."
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Abstract. 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 onedimensional fibers corresponding to the central direction. The sets are genuinely nonhyperbolic 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 ‖ (Ec 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. 1.