Arithmetic and growth of periodic orbits

TL;DR

This paper characterizes when sequences can represent the number of periodic points in dynamical systems, showing limitations for polynomials, conditions for recurrence sequences, and growth rate implications.

Contribution

It provides necessary and sufficient conditions for sequence realizability as periodic point counts, advancing understanding of dynamical system behaviors.

Findings

No non-constant polynomial is realizable as periodic point counts.

Rapidly-growing sequences can always be realized in rate.

Slow-growing sequences cannot be realized in rate.

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.