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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