Thermodynamic Formalism for a family of cellular automata and duality with the shift
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| Format: | Preprint |
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2024
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| _version_ | 1866909243650605056 |
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| author | Lopes, Artur O. Oliveira, Elismar R. Sobottka, Marcelo |
| author_facet | Lopes, Artur O. Oliveira, Elismar R. Sobottka, Marcelo |
| contents | We will consider a family of cellular automata $Φ: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $σ$-invariant and $Φ$-invariant, where $σ$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,σ}$ and $\mathcal{L}_{B,Φ}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $σ$ and $Φ$. Via the use of a version of the involution kernel $W$ for a $(σ,Φ)$-mixed skew product $\hatΦ: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $ρ_A$ for $(\mathcal{L}_{A,σ})^*$ and eigenfunctions $ψ_B$ for $\mathcal{L}_{B,Φ}$. In another direction, considering the non-mixed extension $\hatΦ_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $Φ$, given a Lispchitz potential $\hat{A} : \{1,2,...,r\}^\mathbb{Z}\to \mathbb{R}$, we can identify a Lipschitz potential $A:\{1,2,...,r\}^\mathbb{N} \to \mathbb{R} $, in such away that relates the variational problem of $\hatΦ_n$-Topological Pressure for $\hat{A}$ with the $Φ$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $Φ$ (or $\hatΦ)$ can eventually be conjugated with another shift of finite type is irrelevant in our context. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thermodynamic Formalism for a family of cellular automata and duality with the shift Lopes, Artur O. Oliveira, Elismar R. Sobottka, Marcelo Dynamical Systems Mathematical Physics Probability Cellular Automata and Lattice Gases 37D35, 37B15, 68Q80 We will consider a family of cellular automata $Φ: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $σ$-invariant and $Φ$-invariant, where $σ$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,σ}$ and $\mathcal{L}_{B,Φ}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $σ$ and $Φ$. Via the use of a version of the involution kernel $W$ for a $(σ,Φ)$-mixed skew product $\hatΦ: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $ρ_A$ for $(\mathcal{L}_{A,σ})^*$ and eigenfunctions $ψ_B$ for $\mathcal{L}_{B,Φ}$. In another direction, considering the non-mixed extension $\hatΦ_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $Φ$, given a Lispchitz potential $\hat{A} : \{1,2,...,r\}^\mathbb{Z}\to \mathbb{R}$, we can identify a Lipschitz potential $A:\{1,2,...,r\}^\mathbb{N} \to \mathbb{R} $, in such away that relates the variational problem of $\hatΦ_n$-Topological Pressure for $\hat{A}$ with the $Φ$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $Φ$ (or $\hatΦ)$ can eventually be conjugated with another shift of finite type is irrelevant in our context. |
| title | Thermodynamic Formalism for a family of cellular automata and duality with the shift |
| topic | Dynamical Systems Mathematical Physics Probability Cellular Automata and Lattice Gases 37D35, 37B15, 68Q80 |
| url | https://arxiv.org/abs/2407.04658 |