A Simple Proof of Sharkovsky's Theorem Revisited

Sharkovsky's theorem [1], [5], [6] claims its place among the gems of dynamical sys tems mainly because of its simple hypotheses and strong conclusion. Loosely speak ing, it states that, if / is a continuous map from a compact interval / into itself that has a period-m point, then / also has a period-^ point whenever m < n in the Sharkovsky's ordering -< of the natural numbers but may not have a period-/: point when k < m. The Sharkovsky's ordering is as follows:

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