Conservation Laws and Invariant Measures in Surjective Cellular Automata

: Kari J, Taati S

: Nazim Fatès, Eric Goles, Alejandro Maass, Iván Rapaport

Publisher: DISCRETE MATHEMATICS THEORETICAL COMPUTER SCIENCE

: 2012

: Discrete Mathematics and Theoretical Computer Science

: Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems

: DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE

: DISCRETE MATH THEOR

: Discrete Mathematics and Theoretical Computer Science

: 113

: 122

: 10

: 1462-7264

: https://inria.hal.science/hal-00654706

We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.