Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911356692725760 |
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| author | Mohammadpour, Reza Quas, Anthony |
| author_facet | Mohammadpour, Reza Quas, Anthony |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$ Mohammadpour, Reza Quas, Anthony Dynamical Systems 37H15, 37D35 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. |
| title | Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$ |
| topic | Dynamical Systems 37H15, 37D35 |
| url | https://arxiv.org/abs/2507.01148 |