# Measure concentrated on the $\omega$-limit set

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?

• 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. Jul 6 at 13:31
• 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. Jul 6 at 13:49
• 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... Jul 6 at 21:42
• 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? :) Jul 6 at 21:51
• Maybe you find this useful arxiv.org/abs/1106.4074 Jul 7 at 1:16