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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 the measure distributed on $\omega(x)$ would 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 the latter case, I'm not asking necessarily that the "$\omega$-measure" is invariant under $F$.

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

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?

Let $(X,F)$ be a dynamical system with $X$ compact and $F$ 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 the measure distributed on $\omega(x)$ would 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 the latter case, I'm not asking necessarily that the "$\omega$-measure" is invariant under $F$.

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

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 the measure distributed on $\omega(x)$ would 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 the latter case, I'm not asking necessarily that the "$\omega$-measure" is invariant under $F$.

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