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.Comment: Revised version with new content; to appear in J.Integer Sequence

Puri, Y.

Ward, T.

English

arXiv

Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, is the property that the sequence asympototically approximates the number of periodic points under some map. In both cases we discuss when a sequence can have that property. For exact realizability, this amounts to examining the range and domain among integer sequences of the paired transformations



Pern =   Σd|nd  Orbd;     Orbd =1/n Σd|nµ(n/d)Perd ORBIT



that move between an arbitrary sequence of non-negative integers Orb counting the orbits of a map and the sequence Per of periodic points for that map. Several examples from the Encyclopedia of Integer Sequences arise in this work, and a table of sequences from the Encyclopedia known or conjectured to be exactly realizable is given

Puri, Y

Ward, T

White Rose Research Online

Arithmetic and growth of periodic orbits

Durham Research Online

Two natural properties of integer sequences are introduced and studied. The first, exact realizability, is the property that the sequence coincides with the number of periodic points under some map. This is shown to impose a strong inner structure on the sequence. The second, realizability in rate, is the property that the sequence asymptotically approximates the number of periodic points under some map. In both cases we discuss when a sequence can have that property. For exact realizability, this amounts to examining the range and domain among integer sequences of the paired transformations that move between an arbitrary sequence of non-negative integers Orb counting the orbits of a map and the sequence Per of periodic points for that map. Several examples from the Encyclopedia of Integer Sequences arise in this work, and a table of sequences from the Encyclopedia known or conjectured to be exactly realizable is given

Puri, Yash

Ward, Thomas

University of East Anglia digital repository

Arithmetic and growth of periodic orbits

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White Rose Research Online

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University of East Anglia digital repository