arXiv2024
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.