Sensitive dependence of Gibbs measures at low temperatures

Abstract

The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

Minor corrections geared to improve the exposition

References in corpus (3)

• Chaotic temperature dependence at zero temperature
• Ergodic optimization, zero temperature limits and the max-plus algebra
• Ground States are generically a Periodic Orbit

Cited by in corpus (12)

• Ergodic optimization in dynamical systems
• Zero-temperature phase diagram for double-well type potentials in the summable variation class
• Sensitive dependence of geometric Gibbs states at positive temperature
• Ground states and periodic orbits for expanding Thurston maps
• Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures
• On the selection of subaction and measure for perturbed potentials
• Zero-Temperature Chaos in Bidimensional Models with Finite-Range Potentials
• Product type potential on the $XY$ model: selection of maximizing probability and a large deviation principle
• A topological classification of locally constant potentials via zero-temperature measures
• On the absence of zero-temperature limit of equilibrium states for finite-range interactions on the lattice ${\mathbb{Z}}^2$
• Robust sensitive dependence of geometric Gibbs states for analytic families of quadratic maps
• Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models