Entropy rate of higher-dimensional cellular automata

TL;DR

This paper introduces the concept of entropy rate for multidimensional cellular automata, demonstrating its invariance, finiteness, and properties similar to one-dimensional cases, with specific calculations for permutative automata.

Contribution

It defines and analyzes the entropy rate for higher-dimensional cellular automata, extending properties known from one-dimensional automata and establishing invariance and computability.

Findings

Entropy rate is invariant under shift-commuting isomorphisms.

Entropy rate of 2D automata is finite and preserves properties of 1D entropy.

Finite entropy rate depends on the number of permutative sites.

Abstract

We introduce the entropy rate of multidimensional cellular automata. This number is invariant under shift-commuting isomorphisms; as opposed to the entropy of such CA, it is always finite. The invariance property and the finiteness of the entropy rate result from basic results about the entropy of partitions of multidimensional cellular automata. We prove several results that show that entropy rate of 2-dimensional automata preserve similar properties of the entropy of one dimensional cellular automata. In particular we establish an inequality which involves the entropy rate, the radius of the cellular automaton and the entropy of the d-dimensional shift. We also compute the entropy rate of permutative bi-dimensional cellular automata and show that the finite value of the entropy rate (like the standard entropy of for one-dimensional CA) depends on the number of permutative sites. Finally we define the topological entropy rate and prove that it is an invariant for topological shift-commuting conjugacy and establish some relations between topological and measure-theoretic entropy rates.