Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$

TL;DR

This paper investigates phase transitions in equilibrium measures for matrix cocycles generated by non-negative matrices, revealing non-uniqueness for negative parameters and a transition to unique, fully supported measures.

Contribution

It demonstrates the existence of a freezing phase transition in matrix cocycles with non-negative matrices, even under strong irreducibility and proximality, which was previously unknown.

Findings

Existence of a first-order phase transition at a critical parameter t_c.

Non-uniqueness of equilibrium measures for t < t_c.

Uniqueness and full support of equilibrium measure for t > t_c.

Abstract

We consider a one-step matrix cocycle generated by a pair of non-negative parabolic matrices and study the equilibrium measures for $t\log \|\mathcal A\|$ as $t$ runs over the reals. We show that there is a freezing first order phase transition at some parameter value $t_c$ so that for $t<t_{c}$ the equilibrium measure is non-unique and supported on the two fixed points, while for $t>t_c$, the equilibrium measure is unique, non-atomic and fully supported. The phase transition closely resembles the classical Hofbauer example. In particular, our example shows that there may be non-unique equilibrium measures for negative $t$ even if the cocycle is strongly irreducible and proximal.