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Arithmetic and growth of periodic orbits

Yash Puri·Thomas Ward
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Abstract

We give necessary and sufficient conditions for a sequence to be exactly realizable as the sequence of numbers of periodic points in a dynamical system. Using these conditions, we show that no non-constant polynomial is realizable, and give some conditions on realizable binary recurrence sequences. Realization in rate is always possible for sufficiently rapidly-growing sequences, and is never possible for slowly-growing sequences. Finally, we discuss the relationship between the growth rate of periodic points and the growth rate of points with specified least period.

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Yash Puri, Thomas Ward. 2001-04-26. Arithmetic and growth of periodic orbits. https://arxiv.org/abs/math/9907003

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