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?