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)=T^{r}A(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.