Israel Journal of Mathematics

, Volume 93, Issue 1, pp 387–398

Residual behavior of induced maps

Article

Abstract

Consider (X,F, μ,T) a Lebesgue probability space and measure preserving invertible map. We call this a dynamical system. For a subsetAF. byTA:AA we mean the induced map,TA(x)=TrA(x)(x) whererA(x)=min{i〉0:Ti(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 ofTA which are residual in this metric. The first theorem is due to Conze.Theorem 1 (Conze):For T ergodic, TA is weakly mixing for a residual set of A.Theorem 2:For T ergodic, 0-entropy and loosely Bernoulli, TA is rank-1, and rigid for a residual set of A.Theorem 3:For T ergodic, positive entropy and loosely Bernoulli, TA is Bernoulli for a residual set of A.Theorem 4:For T ergodic of positive entropy, TA 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.

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Copyright information

© The magnes press 1996

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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