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Let $(X,F)$ be a dynamical system with $X$ a compact metric space and $F: X\to X$ continuous. By $\omega$-limit set of a subset $A\subset X$ I mean:

$$\omega(A):= \bigcap_{n=0}^\infty \left(\overline{\bigcup_{m=n}^\infty F^m(A)}\right), $$ which of course in case $A$ is a single point $x\in X$ reduces to the limit points of the forward $F$-orbit of $x$.

It looks natural to me to define a measure on $\omega(A)$ describing how it is "filled" by the trajectories of the points in $A$. For instance, if $x$ tends to a limit cycle, then the measure distributed on $\omega(x)$ should be the Radon measure concentrated and equidistributed on the points of the cycle, while if $x$ is a transitive point which visits "uniformly" the space $X$, it would be the (normalized) Lebesgue measure on $X$.

Notice that, in general, I'm not asking that the "$\omega$-measure" is invariant under $F$, nor I want to confine myself to the subsets of $X$ having full measure. In fact it would already be nice for me to cover the case in which $A$ is just a point.

I'm quite sure this construction is done somewhere, but I'm unable to find a reference work. Can you please help me?

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    $\begingroup$ I am not sure I understand what the precise question is, but if I do, I would suggest you look for the "ergodic decomposition theorem" which associates to a full measure set of points an ergodic measure that describes how the point distributes. $\endgroup$
    – rpotrie
    Commented Jul 6, 2021 at 13:31
  • $\begingroup$ The EDC is meaningful in the ergodic context indeed, while I'm thinking more to the topological dynamical framework. I'm going to edit the question to make it clearer. $\endgroup$
    – Alessandro Della Corte
    Commented Jul 6, 2021 at 13:49
  • $\begingroup$ I think that the question is still imprecise. For $A=\{x\}$ do you just want a measure supported on $\omega(x)$? That can certainly be done. If you want it to describe more accurately how it distributes I guess empirical measures are the only way to go... $\endgroup$
    – rpotrie
    Commented Jul 6, 2021 at 21:42
  • $\begingroup$ I want that the measure is supported on $\omega(x)$ plus additional conditions ensuring that it describes how densely it is attained. And of course it is imprecise... In fact the real question is: is it worth trying to make this precise or (as I believe) someone already did that? :) $\endgroup$
    – Alessandro Della Corte
    Commented Jul 6, 2021 at 21:51
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    $\begingroup$ Maybe you find this useful arxiv.org/abs/1106.4074 $\endgroup$
    – rpotrie
    Commented Jul 7, 2021 at 1:16

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