Residual behavior of induced maps (Q1912798)

Given a Lebesgue probability space \((X,{\mathcal F}, \mu,T)\) and \(A\in {\mathcal F}\) one can define the induced map \(T_A:A\to A\) by \(T_A(x)= T^{r(x)}(x)\) where \(r(x)= \min \{i>0 \mid T(x)\in A\}\). For ergodic \(T\) and a residual set of \(A\) the authors give a number of results investigating the type of transformation \(T_A\) is. The results depend on the whether the entropy is zero or positive, and whether \(T\) is loosely Bernoulli.