Symmetry-breaking-induced loss of ergodicity in maps of the simplex with inversion symmetry

TL;DR

This paper investigates how symmetry-breaking in certain mathematical maps leads to loss of ergodicity, introducing simplified models that reveal multiple invariant measures and complex dynamics due to symmetry and phase space reduction.

Contribution

It provides a novel approach using inversion-symmetric maps of the simplex to analyze ergodicity loss in coupled systems with symmetry, extending understanding of invariant measures in such maps.

Findings

Multiple asymmetric invariant measures can exist in these maps.

Reduction to lower-dimensional maps helps analyze complex dynamics.

Symmetry-breaking induces non-ergodic behavior in the studied systems.

Abstract

Motivated by proving the loss of ergodicity in expanding systems of piecewise affine coupled maps with arbitrary number of units, all-to-all coupling and inversion symmetry, we provide ad-hoc substitutes - namely inversion-symmetric maps of the simplex with arbitrary number of vertices - that exhibit several asymmetric absolutely continuous invariant measures when their expanding rate is sufficiently small. In a preliminary study, we consider arbitrary maps of the multi-dimensional torus with permutation symmetries. Using these symmetries, we show that the existence of multiple invariant sets of such maps can be obtained from their analogues in some reduced maps of a smaller phase space. For the coupled maps, this reduction yields inversion-symmetric maps of the simplex. The subsequent analysis of these reduced maps show that their systematic dynamics is intractable because some essential features vary with the number of units; hence the substitutes which nonetheless capture the coupled maps common characteristics. The construction itself is based on a simple mechanism for the generation of asymmetric invariant union of polytopes, whose basic principles should extend to a broad range of maps with permutation and inversion symmetries.