Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. Characterizing all the invariant measures of a cellular automaton could be challenging. Nevertheless, we establish a connection between the invariance of Gibbs measures (used in statistical mechanics to describe thermodynamic equil...

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individual...

Let (AZ, F ) be a bipermutative algebraic cellular automaton. We present conditions which force a probability measure which is invariant for the N×Z-action of F and the shift map σ to be the Haar measure on Σ, a closed shift-invariant subgroup of the Abelian compact group AZ. This generalizes simultaneously results of B. Host, A. Maass and S. Mart́ınez [7] and M. Pivato [14]. This result is appl...

A general construction for σ−finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of fn ∗ (λ) will imply the existence of a σ−finite invariant measure for the map f which is absolutely continuous with respect to λ, a measure on the phase space describing the sets of measure zero. Furthermore we will...