Abundant rich phase transitions in step-skew products

• Published in: Nonlinearity Vol. 27; no. 9; pp. 2255 - 2280
• Main Authors: Díaz, L J, Gelfert, K, Rams, M
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.09.2014
• Subjects: Construction; Entropy; Fibers; Fibres; Lyapunov exponents; Nonlinear dynamics; Nonlinearity; partially hyperbolic dynamics; Phase transformations; phase transitions; skew product; thermodynamic formalism; Topology; transitivity
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/27/9/2255

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 ) of the potential (Ec the central direction) defined on Λ has k rich phase transitions. This means that there are parameters t , = 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 with positive entropy and different Birkhoff averages. Each phase transition is associated with a gap in the central Lyapunov spectrum of F on Λ.