Computer-Assisted Proof of Loss of Ergodicity by Symmetry Breaking in Expanding Coupled Maps

Abstract

From a dynamical viewpoint, basic phase transitions of statistical mechanics can be regarded as a loss of ergodicity. While many random interacting particle models exhibiting such transitions at the thermodynamics limit exist, finite-dimensional examples with deterministic dynamics on a chaotic attractor are rare, if at all existent. Here, the dynamics of a family of N coupled expanding circle maps is investigated in a parameter regime where absolutely continuous invariant measures are known to exist. At first, empirical evidence is given of symmetry breaking of the ergodic components upon increase in the coupling strength, suggesting that loss of ergodicity should occur for every integer $$N>2$$. Then, a numerical algorithm is proposed which aims to rigorously construct asymmetric ergodic components of positive Lebesgue measure. Due to the explosive growth of the required computational resources, the algorithm successfully terminates for small values of N only. However, this approach shows that phase transitions should be provable for systems of arbitrary number of particles with erratic dynamics, in a purely deterministic setting, without any reference to random processes.