Classification of some metric automorphisms defined by Standish

Abstract

Let F be a finite set with a probability distribution { P i : i ϵ F} and (Ω F , P) denote the product space of countably many copies of ( F, P). A permutation (bijection) φ of the integers induces an invertible measure preserving transformation T φ on (Ω F , P) given by the equation ( T φ w) i = w φ( j) . Such metric automorphisms we call S-automorphisms. We show in this paper that S-automorphisms are ergodic if and only they are Bernoulli shifts and two ergodic S-automorphisms are isomorphic if and only if their associated permutations are conjugate. We also show that S-automorphisms have discrete spectrum if and only if they have zero entropy and every S-automorphism is either a Bernoulli shift, has discrete spectrum, or is a product of a Bernoulli shift and an automorphism with discrete spectrum. S-automorphism with discrete spectrum are those whose associated permutations contain only cycles of finite length. These automorphisms are studied according to the number of such finite cycles. Those whose permutations have infinitely many finite cycles with unbounded lengths are shown to be antiperiodic and those whose permutations have infinitely many finite cycles of bounded length are periodic with almost no fixed points. An example is given of two automorphisms of this latter type which are isomorphic but whose permutations are not conjugate. A complete isomorphism invariant is given for S-automorphisms whose associated permutations consist of finitely many finite cycles. Using this invariant we show that if φ is either a product of k disjoint cycles of prime power p α , or a single cycle of length pq where p and q are primes, or a product of k disjoint cycles of prime lengths p 1 < p 2 < ··· < p k and if ψ is a permutation such that T ψ and T φ are isomorphic then ψ is conjugate to φ.