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

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 $({\mathcal{L}}_{A,\sigma }{)}^{\ast }$ 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.