Integer sequences counting periodic points
Number Theory
2007-05-23 v1 Dynamical Systems
Abstract
An existing dialogue between number theory and dynamical systems is advanced. A combinatorial device gives necessary and sufficient conditions for a sequence of non-negative integers to count the periodic points in a dynamical system. This is applied to study linear recurrence sequences which count periodic points. Instances where the -parts of an integer sequence themselves count periodic points are studied. The Mersenne sequence provides one example, and the denominators of the Bernoulli numbers provide another. The methods give a dynamical interpretation of many classical congruences such as Euler-Fermat for matrices, and suggest the same for the classical Kummer congruences satisfied by the Bernoulli numbers.
Cite
@article{arxiv.math/0204173,
title = {Integer sequences counting periodic points},
author = {Graham Everest and Yash Puri and Thomas Ward},
journal= {arXiv preprint arXiv:math/0204173},
year = {2007}
}