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Not sure if this is what you're looking for, and I'm not an expert on exact systems, but: your definition of exact implies that $\bigcap_{n \geq 0} T^{-n} \mathcal{B} = \mathcal{N}$, where $\mathcal{B …

I may be misunderstanding the question, but in full generality, the answers are "no," i.e. the $\mu_p$ can fail to converge, and even when they converge to a limit $\mu$, $f(\mu_p)$ can fail to conver …

This is something that has confused me too, but let me try to answer anyway. This question itself is a bit tricky to interpret. The reason is that "exponential mixing" really depends on the metric/top …

Not full answer, but too long for a comment. The paper The structure of mean equicontinuous group actions defines your property under the name "pointwise unique ergodicity." Their Theorem 1.2 doesn't …

As Dan alluded to, there's in fact a quite easy way to find measures of cylinder sets for letters, which can extend to larger words with a bit of effort; I'm quite surprised it wasn't mentioned yet. T …

I think the answer is "no" for silly reasons. Consider $X = \{0,1\}^{[0,1]}$ with the product topology and $T$ the identity. Then for every $x \in X$, the "delta-measure" $\delta_x$ is ergodic. I used …

If I understand right, your function $m$ (for a fixed $n$) takes on only finitely many values, which are all measurable sets. You can then define the partition $\{m^{-1}(i)\}$ and the associated finit …

Reposting some proofs I gave in the comments as a (partial) answer, demonstrating that the answer to Julian's question is "yes" for Bernoulli/i.i.d. systems, and so that it is not resolved by some usu …

I don't know if you're still interested now that the average collapses, but I think that there is such a maximizing invertible ergodic $T$ as long as you assume that your probability space is Lebesgu …

I'm not sure if these examples are generalizable for your purposes (I do symbolic dynamics, and the examples I like the most probably have nothing to do with quantum mechanics...), but: Every aperiod …

This is not a complete answer, but too long for a comment. If you remove unique ergodicity as an assumption, then I think there exist systems where it's not possible to find points where all boundarie …

I think that $D_{m,n}$ is just the set of all $m \times n$ matrices $A$ for all $m,n$. The proof is basically the same as that of Dirichlet approximation, i.e., Pigeonhole Principle. For any $m, n$, a …

I don't think that this system is even automatically measure-preserving (unlike the traditional induced map, which as you note is measure-preserving and ergodic). Just take something silly like $(X, T …

I don't think that it's ever possible to mandate any rate of mixing if you are looking at all measurable sets. Here's a silly example using the full $2$-shift $\{0,1\}^{\mathbb{Z}}$ and the i.i.d. mea …

I know that you were mostly asking about cardinality, but more generally, you can ask about the structure of this space $\mathcal{M}(X,T)$ of invariant measures as a topological space with the weak to …