Thermodynamic Formalism for a family of cellular automata and duality with the shift

TL;DR

This paper develops a thermodynamic formalism for a class of cellular automata, establishing conditions for invariant measures, duality relations via involution kernels, and connections between pressure functions, without requiring conjugacy to shifts of finite type.

Contribution

It introduces a novel thermodynamic framework for non-algebraic cellular automata, including duality and pressure relations, expanding the understanding of invariant measures and potentials.

Findings

Identifies conditions for measures invariant under both shift and cellular automaton

Establishes duality between eigenprobabilities and eigenfunctions via involution kernels

Links variational principles of pressure for extended and original automata

Abstract

We will consider a family of cellular automata $\Phi: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ 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 $\sigma$-invariant and $\Phi$-invariant, where $\sigma$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,\sigma}$ and $\mathcal{L}_{B,\Phi}$ 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 $\sigma$ and $\Phi$. Via the use of a version of the involution kernel $W$ for a $(\sigma,\Phi)$-mixed skew product $\hat{\Phi}: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $\rho_A$ for $(\mathcal{L}_{A,\sigma})^*$ and eigenfunctions $\psi_B$ for $\mathcal{L}_{B,\Phi}$. In another direction, considering the non-mixed extension $\hat{\Phi}_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $\Phi$, 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 $\hat{\Phi}_n$-Topological Pressure for $\hat{A}$ with the $\Phi$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $\Phi$ (or $\hat{\Phi})$ can eventually be conjugated with another shift of finite type is irrelevant in our context.