# zbMATH — the first resource for mathematics

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 $$S\neq T$$.

##### MSC:
 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
Full Text: