Dold sequences, periodic points, and dynamics

Abstract

In this survey we describe how the so-called Dold congruence arises in topology, and how it relates to periodic point counting in dynamical systems.

On theorems of Fermat, Wilson, and Gegenbauer

“…We would like to thank the anonymous referee for his/her invaluable insights and the references [4,23]. These references establish crucial connections between our work and Dold sequences.…”

“…We also thank Professors T. J. Evans and S. András for answering our queries with regard to their works. Finally, we thank the referee for the references [4, 23].…”

Time-changes preserving zeta functions

“…A map h ∈ N N lies in P if and only if it preserves two properties on sequences of non-negative integers (a n ) n 1 : If d|n µ(n/d)a d is nonnegative and divisible by n for all n 1 then d|n µ(n/d)a h(d) is nonnegative (property (S)) and divisible by n (property (D)) for all n 1. We refer to the recent survey [2] for more on this and the context in which the two properties-the 'Dold congruence' (D) and the 'sign condition' (S)-sit. The space N N of functions N → N inherits a natural topology of pointwise convergence, meaning that h 1 and h 2 are close in…”

“…The full shift T on 2 symbols has ζ T (z) = 1 1−2z . The Taylor expansion of log 2 1−2z has some non-integer coefficient, showing that Z is not closed under addition. Similarly, log…”