A ``universal'' dynamical system generated by a continuous map of the interval

In this paper we show that there is a continuous map $f:I\rightarrow I$of the interval such that any $\omega$-limit set $W$ of any continuous map $g:I\rightarrow I$ can be transformed by a homeomorphism $I\rightarrow I$ to an $\omega$-limit set $\tilde W$ of $f$. Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an $\omega$-limit set of $f$.