Time-changes preserving zeta functions

TL;DR

This paper investigates specific time-changes in dynamical systems that preserve zeta functions, identifying a monoid of such transformations and characterizing its structure, revealing only monomials as universally good time-changes.

Contribution

It introduces a monoid of time-changes preserving periodic point counts across all dynamical systems and characterizes the polynomials within this monoid as monomials.

Findings

Only monomials are in the universally good monoid.

The monoid of time-changes is uncountable.

Examples illustrate the variation across different systems.

Abstract

We associate to any dynamical system with finitely many periodic orbits of each length a collection of possible time-changes of the sequence of periodic point counts that preserve the property of counting periodic points. Intersecting over all dynamical systems gives a monoid of time-changes that have this property for all such systems. We show that the only polynomials lying in this `universally good' monoid are the monomials, and that this monoid is uncountable. Examples give some insight into how the structure of the collection of maps varies for different dynamical systems.