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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?

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