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

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