Let $T: X \to X$ a continuous map on a compact set $X\subset \mathbb{R}$. A point $x \in X$ is non-wandering if for any open set $U \ni x$ there exists $n>0$ such that $T^{n}(U)\cap U \ne \varnothing$, then $\Omega(T)$ is a set of non-wandering points. Now define $Q=\bigcap_{i \geq 0} {\rm orb}\, Y_i$ (where $Y_{i}$ are subintervals of $X$ for all i) and finally $S_{\Omega}=Q\cap S_{\Omega}(T)$. Who is the set of cluster points of $S_{\Omega}$?
The author says that , if $y \in S_{\Omega}'$, then exists a sequence of $\{U_{i}\}$ of intervals, where every $U_{i}$ is a component of ${\rm orb}\,Y_{i}~ ( \forall i)$, with the following property: $U_{i} \to y$ but $y \not\in U_{i}$, but I do not understand. Can anyone help me?
Thanks a lot.