Abundant rich phase transitions in step-skew products

作者：Diaz L J^*; Gelfert K; Rams M

来源：Nonlinearity, 2014, 27(9): 2255-2280.

DOI：10.1088/0951-7715/27/9/2255

• thermodynamic formalism
• Lyapunov exponents
• phase transitions
• partially hyperbolic dynamics
• skew product
• transitivity

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 Lambda as above such that the pressure map P(t) = P(t phi) of the potential phi = -log parallel to dF vertical bar E-c parallel to (E-c the central direction) defined on Lambda 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) phi with positive entropy and different Birkhoff averages. Each phase transition is associated with a gap in the central Lyapunov spectrum of F on Lambda.

• 出版日期2014-9