# zbMATH — the first resource for mathematics

On the entropy geometry of cellular automata. (English) Zbl 0672.68025
Let $$<K,L,f>$$ be a cellular automaton, where K is a finite alphabet with at least two elements, L is an n-dimensional lattice, i.e. a free abelian group isomorphic to $${\mathbb{Z}}^ n$$ and f: $$K^ L\to K^ L$$ commutes with translations by lattice elements.
The author introduces an n-dimensional entropy and developes a theory of this concept. In this way one gets a very useful tool to measure the flow of information in the $$(n+1)$$-dimensional “space-time lattice” $${\mathbb{Z}}\times L$$ because the classical topological and measure-theoretic entropies are usually infinite in the higher-dimensional case. The paper is an extensive review of many topological as well as measure-theoretic aspects of the introduced notion and is illustrated by a number of explicit examples of cellular automata. Also some generalizations to commuting maps on an arbitrary compact metric space and to commuting measure preserving transformations are given.
Reviewer: W.Jarczyk

##### MSC:
 68Q80 Cellular automata (computational aspects) 28D20 Entropy and other invariants 54H20 Topological dynamics (MSC2010) 94A15 Information theory (general)