Time-changes preserving zeta functions

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.