Let $(X,\mathcal{X},\mu)$ be a probability measure system, $T:X\to X$ be a $\mu$-preserving isomorphism on $X$. Let $A\in \mathcal{X}$ such that $\mu(\bigcup\_{n\ge 0}T^nA)=1$, and $\mu\_A$ be the conditional measure of $\mu$ on $A$. 


For any point $x\in A$, let $n(x)=\inf\{n\ge1: T^nx\in A\}$ be the first return to $A$ (it is finite for $\mu$-a.e. $x\in A$ by Poincare recurrence theorem). Define the first-return map $T\_A:A\to A$, $x\mapsto T^{n(x)}x$, which preserves $\mu\_A$. 


It is well known that $(X,\mu,T)$ is [ergodic](http://en.wikipedia.org/wiki/Ergodicity) if and only if $(A,\mu\_A,T\_A)$ is ergodic. It seems there is a different story for mixing properties.


1. Assume $(X,\mu,T)$ is (strongly) [mixing](http://en.wikipedia.org/wiki/Mixing_%28mathematics%29#Mixing_in_dynamical_systems). Is $(A,\mu\_A,T\_A)$ also mixing?
2. Assume $(A,\mu\_A,T\_A)$ is mixing. Is $(X,\mu,T)$ also mixing?


If not always true, any sufficient condition or counterexample will be also great.


Thanks!




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I forgot some assumptions on the return-time function, that $n$ is **unbounded** and **aperiodic**.


The aperiodicity condition means that the range of $n$ is complicated--it is not contained in some $p\cdot\mathbb{Z}$.


In particular, this assumption rules out the suspension with constant roof functions.



Mixing property of first return map
Any ergodic transformation induces a mixing transformation: This was first proved by Friedman and Ornstein (Advances in Mathematics 10 (1973), 147-163.
I later prove that you can even induce a transformation with Lebesgue spectrum (see [Annales IHP, 1998](http://archive.numdam.org/ARCHIVE/AIHPB/AIHPB_1998__34_2/AIHPB_1998__34_2_249_0/AIHPB_1998__34_2_249_0.pdf)).


