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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

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