Multiplicative Cellular Automata on Nilpotent Groups: Structure, Entropy, and Asymptotics

TL;DR

This paper studies multiplicative cellular automata on groups over monoids like Z^D, characterizing their structure, entropy, and asymptotic behavior, especially focusing on nonabelian finite groups and their measure convergence properties.

Contribution

It provides a structure theory for multiplicative cellular automata on G^M, characterizes when they are group endomorphisms, and analyzes their entropy and measure convergence.

Findings

MCA are characterized as group endomorphisms under certain conditions.

The structure of G influences the properties of MCA.

The entropy of MCA can be explicitly computed, and measures converge to Haar measure.

Abstract

If M is a monoid (e.g. the lattice Z^D), and G is a finite (nonabelian) group, then G^M is a compact group; a `multiplicative cellular automaton' (MCA) is a continuous transformation F:G^M-->G^M which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of G. We characterize when MCA are group endomorphisms of G^M, and show that MCA on G^M inherit a natural structure theory from the structure of G. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure.