Mixing property of first return map

Question

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 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. 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.

Answer 1

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).

Answer 2

What you're asking amount to a time change: in the induced system, you count time by counting how many times you hit $A$, rather than counting how many times you apply the original dynamics. Mixing properties, as you suggest, are very vulnerable to time changes.

An example of this type is an old theorem of Chacon and Parry (proved in Parry's book `Some topics in ergodic theory').

For a really trivial example, but nonetheless, one that captures the spirit nicely: Take your favourite dynamical system: $S\colon Y\to Y$. Set $X=\{0,1,\ldots,n-1\}\times Y$ and define $T(k,y)=(k+1,y)$ if $k<n-1$ and $T(n-1,y)=(0,S(y))$. If $A=\{(0,y)\colon y\in Y\}$, then $T_A$ is isomorphic to $S$, so if $S$ was (weak-)mixing, so is $T_A$. On the other hand $T$ is obviously not weak-mixing because it factors onto the $n$ point system (just factor onto the first coordinate). This is a counterexample to 2.

I don't have an off the shelf counterexample to 1, but the principle "time changes screw up mixing" will certainly apply.