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][1] 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][2]. 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.

  [1]: http://en.wikipedia.org/wiki/Ergodicity
  [2]: http://en.wikipedia.org/wiki/Mixing_%28mathematics%29#Mixing_in_dynamical_systems