Residual behavior of induced maps

Abstract

Consider (X,F, μ,T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subsetA ∈F. byT A:A →A we mean the induced map,T A(x)=TrA(x)(x) wherer A(x)=min{i〉0:T i(x) ∈A}. Such induced maps can be topologized by the natural metricD(A, A’) = μ(AΔA’) onF mod sets of measure zero. We discuss here ergodic properties ofT A which are residual in this metric. The first theorem is due to Conze.Theorem 1 (Conze):For T ergodic, T A is weakly mixing for a residual set of A.Theorem 2:For T ergodic, 0-entropy and loosely Bernoulli, T A is rank-1, and rigid for a residual set of A.Theorem 3:For T ergodic, positive entropy and loosely Bernoulli, T A is Bernoulli for a residual set of A.Theorem 4:For T ergodic of positive entropy, T A is a K-automorphism for a residual set of A. A strengthening of Theorem 1 asserts thatA can be chosen to lie inside a given factor algebra ofT. We also discuss even Kakutani equivalence analogues of Theorems 1–4.