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Integer sequences counting periodic points

Graham Everest·Yash Puri·Thomas Ward
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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 $p$-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.

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Graham Everest, Yash Puri, Thomas Ward. 2002-04-13. Integer sequences counting periodic points. https://arxiv.org/abs/math/0204173

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