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I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and have run into a question about entropy and Bowen balls.

Setup: $T$ is a homeomorphism from a compact metric space $X$ to itself. Definition: $B(x,n,\delta)=\lbrace y\colon d(T^ix,T^iy) < \delta\text{ for $0\le i < n$}\rbrace$ is a Bowen $(n,\delta)$-ball around $x$. Let $\mu$ be an ergodic invariant measure for $T$ and let $A$ be a set of positive measure.

What can be said about the functions $$ f_n(x)=\frac{\mu(A\cap B(x,n,\delta))}{\mu(B(x,n,\delta))} $$ as $n\to\infty$ for fixed $\delta$?

Notice that if $X$ is a shift space, then the $B(x,n,\delta)$ partition the space and the $f_n$ are just conditional expectations with respect to that partition. In that case, $\int f_n \ d\mu(x)$ is equal to $\mu(A)$ for all $n$. This is the kind of conclusion I'd like to find in the general case.

One interpretation of the question is that I'm asking: if you pick a point $x$ according to the measure $\mu$ and then pick a second point $y$ according to the restriction of $\mu$ to the Bowen ball around $x$, then is the distribution of $y$ `similar' to $\mu$?

For a non-dynamical example where the resampled distribution is not the same, consider the set $\lbrace 1,2,3\rbrace$ with the usual distance in $\mathbb R$. If you pick a point $x$, and then sample a point $y$ in the 1.5 ball around $x$, you're more likely to get to 2 than you are to 1 or 3.

If anyone has ideas, or has seen something similar, I'd really like to hear about it...

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