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!

I forgot to add an important assumption that the return-time function $n$ is unbounded. In particular, this assumption rules out the suspension with constant roof functions.

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!

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.

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!

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!

I forgot to add an important assumption that the return-time function $n$ is unbounded. In particular, this assumption rules out the suspension with constant roof functions.