Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative t
Reza Mohammadpour, Anthony Quas · Nonlinearity · 2026
Abstract We consider the one-step matrix cocycle generated by a particular pair of non-negative parabolic matrices and study the equilibrium measures for t log ‖ A ‖ as t runs over the reals. We show that there is a freezing first order phase transition at t = − 2 so that for t ⩽ − 2 the equilibrium measure is non-unique and supported on the two fixed points, while for t > − 2 , 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.