A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line
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Two theorems are proved—the first and the more important of them due to Šarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.
KeywordsEntropy Neural Network Statistical Physic Complex System Lower Bound
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