On a Constraint on Invariant Measures of Certain Cellular Automata

TL;DR

This paper explores constraints on invariant measures of certain cellular automata, linking group structure to measure properties, and extends the analysis to a broader class called RLP subshifts.

Contribution

It strengthens previous formulations of measure constraints, investigates their implications, and generalizes results to RLP subshifts beyond bi-permutative automata.

Findings

Invariant measures are determined by fixed values at positive indices.

In finite group cases, invariant measures relate to cosets of subgroups.

Zero entropy measures on certain factors relate to positive entropy measures on the original system.

Abstract

In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index. When the alphabet is a finite group and the automaton is multiplication of two neighbors, that set is in fact a coset of some subgroup. In the present paper, we strengthen the formulations in [6] and investigate further the implications of this constraint. In the finite group case mentioned above, relations between some attributes of the group structure and the invariant measures are examined. We also inspect a factor, with respect to the shift, that this constraint induces, and analyze the special case in which it has zero measuretheoretical entropy, thus observing an interplay between zero entropy invariant measures on that factor and the positive entropy measures corresponding to them on the original system. Then, we leave the setting of bi-permutative cellular automata and generalize our results to a wider class which we named RLP subshifts. The peculiar situation is that although we conjecture that this class is much larger than the class of bi-permutative cellular automata, we were able to prove only for essentially one other example that it belongs to it.