Let $f: X\to X$ be a measurable map of a measurable space $X$.

Define a measure $\mu$ to be {\bf absolutely equicontinuous w.r.t. $f$}

if for every $\epsilon>0$ there is $\delta>0$ such that

$\mu(A)<\delta$ implies $\mu(f^{-m}(A))<\epsilon$ for all $m\in \mathbb{N}$.

For example, every invariant measure is equicontinuous w.r.t. $f$.

My question is:

Are there expansive homeomorphisms $f$ on decent compact metric spaces

(e.g. ones without isolated points) exhibiting NON-INVARIANT Borel probability measures

which are absolutely equicontinuous w.r.t. $f$?

Thanks for watching!

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In fact, that’s exactly what makes the representation theory of (the group algebra of) interesting while the representation theory of a field is boring: the only interesting thing about a field representation (vector space) is its dimension — the number of generators — and each generator behaves the same as every other. -modules have a lot more structure than that.

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