Mean dimension of continuous cellular automata
David Burguet, Ruxi Shi
TL;DR
This paper studies the mean dimension of continuous cellular automata, providing formulas for certain classes and examples illustrating infinite and finite cases across different dimensions.
Contribution
It establishes a formula for mean dimension in specific classes of cellular automata and presents examples with finite and infinite mean dimension.
Findings
Mean dimension formula for strongly permutative and algebraic automata
Higher-dimensional permutative automata have infinite mean dimension
Existence of algebraic surjective automata with positive finite mean dimension
Abstract
We investigate the mean dimension of a cellular automaton (CA for short) with a compact non-discrete space of states. A formula for the mean dimension is established for (near) strongly permutative, permutative algebraic and unit one-dimensional automata. In higher dimensions, a CA permutative algebraic or having a spaceship has infinite mean dimension. However, building on Meyerovitch's example, we give an example of algebraic surjective cellular automaton with positive finite mean dimension.
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Taxonomy
TopicsCellular Automata and Applications · Advanced Operator Algebra Research · Markov Chains and Monte Carlo Methods