Large deviation for Gibbs probabilities at zero temperature and invariant idempotent probabilities for iterated function systems
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| Format: | Preprint |
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2024
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| author | Mengue, Jairo. K. Oliveira, Elismar R. |
| author_facet | Mengue, Jairo. K. Oliveira, Elismar R. |
| contents | We consider two compact metric spaces $J$ and $X$ and a uniform contractible iterated function system $\{ϕ_j: X \to X \, | \, j \in J \}$. For a Lipschitz continuous function $A$ on $J \times X$ and for each $β>0$ we consider the Gibbs probability $ρ_{_{βA}}$. Our goal is to study a large deviation principle for such family of probabilities as $β\to +\infty$ and its connections with idempotent probabilities. In the non-place dependent case ($A(j,x)=A_j,\,\forall x\in X$) we will prove that $(ρ_{_{βA}})$ satisfy a LDP and $-I$ (where $I$ is the deviation function) is the density of the unique invariant idempotent probability for a mpIFS associated to $A$. In the place dependent case, we prove that, if $(ρ_{_{βA}})$ satisfy a LDP, then $-I$ is the density of an invariant idempotent probability. Such idempotent probabilities were recently characterized through the Mañé potential and Aubry set, therefore we will obtain an identical characterization for $-I$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviation for Gibbs probabilities at zero temperature and invariant idempotent probabilities for iterated function systems Mengue, Jairo. K. Oliveira, Elismar R. Dynamical Systems Mathematical Physics 37A30, 37A50, 28A33, 46E27, 60B10, 60F10, 15A80, 37C30 We consider two compact metric spaces $J$ and $X$ and a uniform contractible iterated function system $\{ϕ_j: X \to X \, | \, j \in J \}$. For a Lipschitz continuous function $A$ on $J \times X$ and for each $β>0$ we consider the Gibbs probability $ρ_{_{βA}}$. Our goal is to study a large deviation principle for such family of probabilities as $β\to +\infty$ and its connections with idempotent probabilities. In the non-place dependent case ($A(j,x)=A_j,\,\forall x\in X$) we will prove that $(ρ_{_{βA}})$ satisfy a LDP and $-I$ (where $I$ is the deviation function) is the density of the unique invariant idempotent probability for a mpIFS associated to $A$. In the place dependent case, we prove that, if $(ρ_{_{βA}})$ satisfy a LDP, then $-I$ is the density of an invariant idempotent probability. Such idempotent probabilities were recently characterized through the Mañé potential and Aubry set, therefore we will obtain an identical characterization for $-I$. |
| title | Large deviation for Gibbs probabilities at zero temperature and invariant idempotent probabilities for iterated function systems |
| topic | Dynamical Systems Mathematical Physics 37A30, 37A50, 28A33, 46E27, 60B10, 60F10, 15A80, 37C30 |
| url | https://arxiv.org/abs/2405.12793 |