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An uncountable family of group automorphisms, and a typical member. (English) Zbl 0903.22001
If \(\alpha: X\to X\) is an isomorphism of a compact abelian group, then \(\alpha\) is ergodic if and only if it is measurably isomorphic to a Bernoulli shift by [D. A. Lind, Israel J. Math. 28, 205–248 (1977; Zbl 0365.28015)]. In this note it is shown by giving examples that the equivalence relation of topological conjugacy of compact group automorphisms with prescribed entropy \(\log 2\) has uncountably many equivalence classes and that uncountably many of them can be distinguished by their dynamical zeta function. The examples are constructed as projective limit of circle groups \(\mathbb R/ \mathbb Z\) with respect to multiplication by natural numbers having prime divisors only in a given set of primes \(S\). Every set \(S\) thus gives a compact group \(X_S\) and if we insist that \(S\) should contain the prime 2 we get an automorphism \(\alpha_S\) defined by the multiplication by 2. The so defined dynamical system has entropy \(\log 2\) and each \(\alpha_S\) is measurably isomorphic to a Bernoulli shift, but different sets \(S\) give different zeta functions and these zeta functions are generically irrational. Further \(\alpha_S\) and \(\alpha_T\) are not topologically conjugate if \(S\neq T\).

37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37B99 Topological dynamics
22D40 Ergodic theory on groups
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
28D05 Measure-preserving transformations
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