Module Shifts and Measure Rigidity in Linear Cellular Automata

TL;DR

This paper investigates the structure of invariant measures in linear cellular automata over finite modules, showing that under certain conditions, such measures are necessarily uniform or Haar measures, revealing rigidity phenomena.

Contribution

It establishes measure rigidity results for linear cellular automata over finite modules, linking invariance, mixing properties, and measure classification.

Findings

Invariant measures are Haar measures on cosets of submodule shifts.

Under certain conditions, invariant measures are the uniform Bernoulli measure.

The results demonstrate measure rigidity in linear cellular automata.

Abstract

Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.