Sensitive dependence of geometric Gibbs states at positive temperature

Abstract

We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature.

Minor changes. arXiv admin note: substantial text overlap with arXiv:1708.03965, arXiv:1305.4971

References in corpus (2)

• Chaotic temperature dependence at zero temperature
• Zero-temperature phase diagram for double-well type potentials in the summable variation class

Cited by in corpus (3)

• Ground states and periodic orbits for expanding Thurston maps
• Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models
• Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps