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arXiv · 2507.01148

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

Reza Mohammadpour·Anthony Quas
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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 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.

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Reza Mohammadpour, Anthony Quas. 2025-07-01. Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$. https://arxiv.org/abs/2507.01148

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