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An algebraic obstruction to isomorphism of Markov shifts with group alphabets. (English) Zbl 0792.22004

Given a compact group \(G\), a standard construction of a \(\mathbb Z^ 2\) Markov shift \(\Sigma_ G\) with alphabet \(G\) is described. The cardinality of \(G\) (if \(G\) is finite) or the topological dimension of \(G\) (if \(G\) is a torus) is shown to be an invariant of measurable isomorphism for \(\Sigma_ G\). We show that if \(G\) is sufficiently nonabelian (for instance \(A_ 5\), \(\text{PSL}_ 2(\mathbb F_ 7)\) or a Suzuki simple group) and \(H\) is any abelian group with \(| H| = | G|\), then \(\Sigma_ G\) and \(\Sigma_ H\) are not isomorphic. Thus the cardinality of \(G\) is seen to be necessary but not sufficient to determine the measurable structure of \(\Sigma_ G\).

MSC:

22D40 Ergodic theory on groups
28D20 Entropy and other invariants
43A05 Measures on groups and semigroups, etc.
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