Time-changes preserving zeta functions

Abstract

We associate to any dynamical system with finitely many periodic orbits of each period a collection of possible time-changes of the sequence of periodic point counts for that specific system that preserve the property of counting periodic points for some system. 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 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.

Dold sequences, periodic points, and dynamics

“…Using the closure under infinite composition property mentioned above, this allows us to embed the power set of the set of primes into P. One of the questions raised in [58] asked if a non-trivial permutation could lie in P, and we answer this here, in a stronger form.…”

“…• Symmetry: As discussed in Section 4.4, there is a notion of 'symmetry' in the space of all zeta functions that shows, for example, if (a n ) is a realisable sequence then (a n 2 ) is also a realisable sequence. These are 'symmetries' of the space of realisable (and hence of Dold) sequences that clearly do not preserve the Lefschetz property (see Jaidee et al [58]).…”

“…In the dynamical context relations like (3) may be used to prove integrality of certain sequences; we refer to Jaidee et al [58], for example. By Möbius inversion we may deduce from the definition of B a classical relation due to Möbius [91],…”

“…We can interpret this observation by saying that the map n → 2n for n ∈ N is a realisability-preserving time-change in the following sense. Definition 4.13 (Jaidee et al [58]). For a system (X, T ) define P T , the set of realisability-preserving time-changes for (X, T ), to be the set of maps h : N → N with the property that (F T (h(n))) is a realisable sequence.…”

Dold sequences, periodic points, and dynamics

“…This proves that c ′ n = c n for all n 1, completing the proof. In the dynamical context relations like (3) may be used to prove integrality of certain sequences; we refer to Jaidee et al [45] for examples. By Möbius inversion we may deduce from the definition of B a classical relation due to Möbius [69],…”