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It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already answered in the literature? or is it just too simple?

Most of my search results talk about the measure that's the result of an ergodic system, and not the other way around. Could somebody shed light on this please?

EDIT: Based on the comment of @WillSawin I am adding that, let's assume the distribution is defined over $\mathbb{R}^n$, or maybe more generally over a Riemannean manifold, with a compact support. Additionally the dynamical system has a flow which is a smooth function of time.

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    $\begingroup$ What kind of measure spaces are you looking for distributions on? For many notions of "distribution" and "nice" there is a theorem of the form "there is a single distribution that every nice distribution is equivalent to" so the result you're looking for would be true, but basically trivial. $\endgroup$
    – Will Sawin
    Sep 10, 2021 at 19:56
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    $\begingroup$ If the support has two disjoint components then the flow I don't think the flow can be a smooth function of time and also be ergodic since the flow would not be able to mix the two components. So there must be some criteria. $\endgroup$
    – Will Sawin
    Sep 10, 2021 at 22:07
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    $\begingroup$ Instead of looking at flows, maybe it would be simpler to ask about a discrete dynamical system generated by a continuous map? $\endgroup$
    – Martin M. W.
    Sep 10, 2021 at 22:26
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    $\begingroup$ For the uniform measure on $S^2$, you may run in trouble finding a continuous time dynamical system it is ergodic for, because of the hairy ball theorem. $\endgroup$
    – Will Sawin
    Sep 13, 2021 at 19:35
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    $\begingroup$ Slowly raising from the north pole to the south pole doesn't sound like it has an invariant measure supported on the whole space, not to mention being ergodic. $\endgroup$
    – Will Sawin
    Sep 13, 2021 at 20:12

1 Answer 1

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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 orbit of $P$.)

This distribution seems like it should count as "well-behaved." Its support is connected, and both $\mu$ and $\delta_P$ themselves arise as time-averages (by a rotational flow, and a flow to an attracting fixed point, respectively).

You can generalize to $\mathbb{R}^n$ by looking at an embedded $S^1$, and generalize to continuous maps by replacing the rotational flow with an irrational rotation.

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  • $\begingroup$ Thank you, great example. Do you mind if I update my question by asking that for the given measure with support $M$, and $\delta>0$ there exist another measure such as $\mu'$, where $\mu'$ with support $M$ which is the time average of a dynamical system, and additionally $\|\mu-\mu'\|_{KS}<\delta$. I feel your example is in line with how one makes a counter-example to convert Boltzmannean ergodicity to von Neumannean ergodicity. No? $\endgroup$
    – Cupitor
    Sep 13, 2021 at 18:53
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    $\begingroup$ It might be simplest to ask a new question, and make reference to this one as motivation. But feel free to add an update if you want (it's your question!), just make it clear what the old version of the question was, so this answer and other comments still make sense. $\endgroup$
    – Martin M. W.
    Sep 13, 2021 at 21:03

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