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For the specific case you mention of an irrational rotation of the circle, it depends on the rotation number $r$. You can get slow asymptotic convergence for something like $$r=\sum{10^{-n!}}$$ (and y …

You can't always approximate by ergodic flows, because ergodic flows might not even exist. For example, on $S^2$ the Poincare-Bendixson theorem rules out ergodic flows, but there are many measure-pres …

Unless I'm misinterpreting the question, the SRB measure is just Lebesgue measure... the cat map is hyperbolic, preserves area, and is topologically transitive. See Theorem 3.10 and the following rema …

The reason that the matrices are alike is that, up to a multiplicative factor, each is approximately encoding $Area(T(R_i) \cap R_j))$. This is because the rational points with denominator $q$ are clo …

Let $\mu$ be Lebesgue measure on $S^1$, and $\delta_P$ be a point-mass at a point $P \in S^1$. Then there is no flow on $S^1$ whose time averages lead to $\frac{1}{2}(\mu + \delta_P)$. (Consider the o …

I think this runs into trouble at condition 1. For example, let $f : M \to M$ be an Anosov diffeomorphism that satisfies these conditions. The map $g: M \times S^1 \to M \times S^1$ defined by $g = f …