标题： 非唯一的平衡测度和负的$t$的矩阵上同调的冻结相变 显示英文标题

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

摘要： 我们考虑由一对非负抛物矩阵生成的一步矩阵上同调，并研究当$t$在实数范围内变化时，$t\log \|\mathcal A\|$的平衡测度。 我们证明在某个参数值$t_c$处存在冻结的一阶相变，使得当$t<t_{c}$时，平衡测度不唯一且支持于两个不动点，而当$t>t_c$时，平衡测度是唯一的、非原子的且完全支持的。 相变与经典的 Hofbauer 示例非常相似。 特别是，我们的示例表明，即使上同调是强不可约且逼近的，对于负的$t$也可能存在不唯一的平衡测度。 显示英文摘要

摘要： 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.