Phase diagram for intermittent maps
Daniel Coronel, Juan Rivera-Letelier
TL;DR
The work analyzes phase diagrams for Hölder potentials under the Pomeau–Manneville intermittent map, revealing two dynamical phases—an intermittent phase with a unique Gibbs state and a stationary phase—separated by a phase transition locus where the pressure loses real-analyticity. It develops a rigorous geometric and analytic description of the phase-transition locus as a co-dimension-1 real-analytic submanifold and links phase transitions to ground states and to persistence phenomena in temperature. A central contribution is the complete characterization of how phase transitions in temperature arise, persist, or fail, including a detailed treatment of the geometric potential and a rigidity result when γ=α. These results connect thermodynamic formalism with the fine structure of intermittent dynamics, providing tools to understand low-temperature limits and ground-state configurations in nonuniformly expanding systems. The findings have implications for quantifying how qualitative dynamical behavior changes with smooth perturbations of the potential and for interpreting phase transitions in related statistical-mechanical models.
Abstract
We explore the phase diagram for potentials in the space of Hölder continuous functions of a given exponent and for the dynamical system generated by a Pomeau--Manneville, or intermittent, map. There is always a phase where the unique Gibbs state exhibits intermittent behavior. It is the only phase for a specific range of values of the Hölder exponent. For the remaining values of the Hölder exponent, a second phase with stationary behavior emerges. In this case, a co-dimension 1 submanifold separates the intermittent and stationary phases. It coincides with the set of potentials at which the pressure function fails to be real-analytic. We also describe the relationship between the phase transition locus, (persistent) phase transitions in temperature, and ground states.