On Sets of Periodic Orbit Lengths in Finitely Presented Dynamical Systems
Huub de Jong
TL;DR
The paper addresses the problem of classifying period sets and least-period sets for finitely presented dynamical systems, including SFTs and sofic shifts. It leverages rationality of the derivative of the dynamical zeta function to obtain linear recurrences for the period counts, and applies the Skolem–Mahler–Lech theorem to derive a finite-plus-arithmetic-progressions structure for period sets. For FP systems, it proves a comprehensive LPS classification: LPS$(f)=F\cup\bigcup d_i S_i$ with $F$ finite and each $S_i$ cofinite; in the irreducible SFT and mixing cases, sharper forms emerge (singletons or cofinite). The paper also provides detailed realizations, via SFTs, sofic shifts, and gap shifts, showing that all identified LPS patterns can be realized by explicit symbolic systems, thereby linking abstract structure to concrete models and informing embedding-type results in symbolic dynamics.
Abstract
We classify the sets of natural numbers $n$ for which certain dynamical systems $(X,f)$ on a compact metric space $X$ have a periodic point of (least) period $n$. Interest in this question dates back to Sharkovskii's theorem for continuous maps on intervals of the real line, but it also ties to checkable conditions for Krieger's embedding theorem for symbolic dynamical systems. Given a system for which the logarithmic derivative of the Artin-Mazur zeta function is rational, we use the Skolem-Mahler-Lech theorem to classify for which $n$ the system has a periodic point of (not necessarily least) period $n$. Moreover, we build on work on finitely presented (FP) systems and their relationship to symbolic dynamics to classify the set of least periods, that is periodic orbit lengths, for arbitrary FP systems, extending a known classification for shifts of finite type. We also provide several constructions to realize any such least period sets.