  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.
