Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward map
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2024
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| _version_ | 1866911818542219264 |
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| author | Lopes, A. O. Oliveira, E. R. |
| author_facet | Lopes, A. O. Oliveira, E. R. |
| contents | We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $Ω=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $Π_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hatΠ\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hatΠ\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hatΠ$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_19566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward map Lopes, A. O. Oliveira, E. R. Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37D35 We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $Ω=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $Π_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hatΠ\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hatΠ\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hatΠ$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case. |
| title | Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward map |
| topic | Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37D35 |
| url | https://arxiv.org/abs/2403.19566 |