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

2025/07/01 by Reza Mohammadpour, Mohammadpour, Reza, Anthony Quas +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #37D35 #37H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2507.01148

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a one-step matrix cocycle generated by a pair of non-negative parabolic matrices and study the equilibrium measures for tlog ‖\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 tt_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.

Citations

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• Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications

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