Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative <i>t</i>

Abstract We consider the one-step matrix cocycle generated by a particular pair of non-negative parabolic matrices and study the equilibrium measures for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">A</mml:mi> </mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:mrow> </mml:math> as t runs over the reals. We show that there is a freezing first order phase transition at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> so that for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mtext>⩽</mml:mtext> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> the equilibrium measure is non-unique and supported on the two fixed points, while for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , 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.