Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. 
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define the sets 
$$A\_N=\{x\in X: T^n(x)\in A \forall |n|<N\}$$


By the Birkhoff ergodic theorem $\mu(A\_N)\to 0$ as $N\to \infty$.


I am looking for a mixing condition on $T$ that will guarantee the following:


For any measurable set $A\subset X$ with $0<\mu(A)<1$ the measure $\mu(A\_N)$ decays exponentially in $N$.


Is this true for all mixing systems?


If not, will this be true for instance when $(X,\mu,T)$ is a Kolmogorov system?


When it is exponentially mixing?



Is there a mixing condition to get the decay property I want?
The desired result is false for all mixing systems other than a point:


**Proposition:** let $T$ be an invertible totally ergodic transformation of a standard probability space $(X,\mathcal{F},\mu)$. Then there exists a measurable set $A\subset X$ such that $0<\mu(A)<1$ and


$$\mu\left(\left\{x \in X \colon T^nx \in A\text{ for all } n \in [-N,N]\right\}\right)\geq \frac{1}{2N^3+2}$$


for all $N>1$.


The proof modifies an argument of G. Hal\'asz [Remarks on the remainder in the Birkhoff ergodic theorem, *Acta Mathematica Academiae Scientiarum Hungarica* 28 (1976) 389-395]. The lower bound given above is somewhat arbitrary and can be modified at the expense of brevity. We use the following consequence of Rokhlin's Lemma: for every $\varepsilon>0$ and $N \geq 1$, we may find a set $B\in\mathcal{F}$ such that the sets $T^iB$ are disjoint for $|i|\leq N$, and such that their total measure is at least $1-\varepsilon$.


For each $N \geq 2$, let us choose a set $B\_N$ such that the sets $T^iB\_N$ are disjoint for $|i|\leq N$, and such that their union has measure at least $\frac{2N+1}{2N+2}$. In particular $\frac{1}{2N+2}\leq \mu(B\_N)\leq \frac{1}{2N+1}$ since $X$ is a probability space and $T$ preserves measure. Now define
$$A:=\bigcup\_{N=2}^\infty \bigcup\_{|i|\leq N} T^iB\_{N^3}.$$
The measure of $A$ is bounded below by $\mu(B\_{2^3})>0$ and above by
$$\sum\_{N=2}^\infty (2N+1)\mu(B\_{N^3})\leq \sum\_{N=2}^\infty \frac{2N+1}{2N^3+1}<\sum\_{N=2}^\infty \frac{3N}{2N^3}=\frac{\pi^2}{4}-\frac{3}{2}<1.$$
Now note that if $x \in B\_{N^3}$ then $T^ix \in T^iB\_{N^3}\subset A$ when $|i|\leq N$, and therefore
$$\mu\left(\left\{x \in X \colon T^nx \in A\text{ for all } n \in [-N,N]\right\}\right)\geq \mu(B\_{N^3})\geq \frac{1}{2N^3+2}$$
as claimed.


