On Sets of Periodic Orbit Lengths in Finitely Presented Dynamical Systems

Huub de Jong

arxiv: 2510.10848 · v3 · submitted 2025-10-12 · 🧮 math.DS

On Sets of Periodic Orbit Lengths in Finitely Presented Dynamical Systems

Huub de Jong This is my paper

Pith reviewed 2026-05-18 07:44 UTC · model grok-4.3

classification 🧮 math.DS

keywords periodic orbitsleast periodsfinitely presented systemsArtin-Mazur zeta functionSkolem-Mahler-Lech theoremsymbolic dynamics

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The pith

Finitely presented dynamical systems have their sets of periodic orbit lengths classified by the Skolem-Mahler-Lech theorem when the logarithmic derivative of the Artin-Mazur zeta function is rational, and every such set is realizable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper classifies the natural numbers n for which dynamical systems on compact metric spaces have periodic points of period n or least period n. For systems satisfying the rationality condition on the logarithmic derivative of the Artin-Mazur zeta function, the Skolem-Mahler-Lech theorem determines exactly which n occur as periods. For arbitrary finitely presented systems, the possible sets of least periods are classified and shown to be exactly those realizable by some finitely presented system. This extends the classification previously known only for shifts of finite type. A sympathetic reader would care because the result supplies a concrete, checkable description of allowed period sets that connects to embedding problems in symbolic dynamics.

Core claim

For a system whose logarithmic derivative of the Artin-Mazur zeta function is rational, the Skolem-Mahler-Lech theorem classifies the n for which a periodic point of period n exists. For arbitrary finitely presented systems the sets of least periods are classified and every such set is realized by some finitely presented system.

What carries the argument

the rationality condition on the logarithmic derivative of the Artin-Mazur zeta function, which lets the Skolem-Mahler-Lech theorem classify the periods via the recurrence satisfied by the periodic-point counts.

If this is right

The set of all periods is a finite union of arithmetic progressions, possibly after removing a finite set.
Every set of least periods realizable by a shift of finite type is also realizable by some finitely presented system.
Explicit constructions exist that produce a finitely presented system realizing any allowed set of least periods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The classification supplies a practical test for whether a given set of periods can arise in a finitely presented system, which may help decide embeddability questions beyond shifts of finite type.
One could test whether the same period-set families appear in broader classes of systems that are not finitely presented but still have rational zeta derivatives.

Load-bearing premise

The dynamical systems under study are finitely presented or satisfy the rationality condition on the logarithmic derivative of the Artin-Mazur zeta function.

What would settle it

An explicit finitely presented system whose set of least periods lies outside the family described by the classification, or for which no finitely presented realization exists, would falsify the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends SFT least-period classification to FP systems with explicit realizations, using Skolem-Mahler-Lech under zeta rationality.

read the letter

Dear colleague, The one or two things to know about this paper are that it classifies the sets of natural numbers n for which finitely presented dynamical systems have periodic points of least period n, extending the known results for shifts of finite type, and it provides several constructions to realize any such admissible set. Additionally, for systems where the logarithmic derivative of the Artin-Mazur zeta function is rational, it applies the Skolem-Mahler-Lech theorem to classify the n with periodic points of any period. The paper does a good job building on existing theory for FP systems and their relation to symbolic dynamics. The constructions are the part that adds value, as they show realizability explicitly rather than leaving it abstract. This could be helpful for problems involving embeddings or understanding periodic behavior in a wider class of systems than just SFTs. The use of the number-theoretic theorem is standard and fits the rationality assumption stated upfront. Where it might be soft is in the details of verifying that the constructions indeed produce finitely presented systems without introducing extra periods or violating the conditions. The abstract mentions building on prior work, so the novelty is in the extension and the realizations. If the proofs are complete and handle the edge cases properly, the central claims should stand. There doesn't appear to be any load-bearing flaw or circular reasoning from what's described. This kind of paper is for specialists in symbolic dynamics and ergodic theory who are interested in classifications of periodic orbits and their applications to embedding theorems like Krieger's. A reader with background in these areas would find the explicit constructions useful for further work. It seems coherent and engaged with the literature, so it deserves a serious referee to check the technical details. I would recommend putting it through peer review.

Referee Report

1 major / 2 minor

Summary. The paper classifies sets of natural numbers n for which dynamical systems (X,f) on compact metric spaces have periodic points of (least) period n. For systems where the logarithmic derivative of the Artin-Mazur zeta function is rational, the Skolem-Mahler-Lech theorem is invoked to classify n with periodic points of period n. For arbitrary finitely presented (FP) systems, the sets of least periods are classified by extending the known classification for shifts of finite type (SFTs), with explicit constructions realizing every set in the class.

Significance. If the derivations hold, the work provides a structural classification of periodic orbit length sets in FP systems, linking symbolic dynamics to the Skolem-Mahler-Lech theorem via generating functions. The explicit constructions demonstrate that the classified sets are sharp and realizable, strengthening the extension beyond SFTs. This offers potential checkable conditions relevant to embedding theorems and builds on existing FP-system theory with machine-checkable potential in the number-theoretic step.

major comments (1)

[§4] §4 (classification of least periods): the extension from SFTs to general FP systems relies on the rationality condition for the zeta-function derivative in the first part; it is unclear whether the structural classification for least periods remains effective or complete when the rationality assumption is dropped, which is load-bearing for the claim that 'any such set is realizable by some FP system'.

minor comments (2)

[Abstract, §2] Abstract and §2: the distinction between 'periodic point of period n' and 'least period n' is introduced but the notation for Fix(f^n) vs. the least-period counting function could be clarified with an explicit example early in the text.
The paper invokes Skolem-Mahler-Lech under the stated rationality condition; a brief remark on the effective computability of the resulting period sets (given the recurrence) would aid readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive recommendation, and the insightful comment on §4. We address the major comment below with a clarification of the logical structure of the paper and indicate the revision that will be made.

read point-by-point responses

Referee: [§4] §4 (classification of least periods): the extension from SFTs to general FP systems relies on the rationality condition for the zeta-function derivative in the first part; it is unclear whether the structural classification for least periods remains effective or complete when the rationality assumption is dropped, which is load-bearing for the claim that 'any such set is realizable by some FP system'.

Authors: We appreciate the referee drawing attention to this point. The two main results of the paper are logically independent. The application of the Skolem-Mahler-Lech theorem in the first part requires the rationality of the logarithmic derivative of the Artin-Mazur zeta function and classifies the set of (not necessarily least) periods for such systems. In contrast, the classification of sets of least periods for arbitrary finitely presented systems in §4 extends the known SFT classification using only the general theory of FP systems (as developed in the cited works on FP dynamical systems) and does not invoke rationality of the zeta-function derivative at any step. The explicit constructions realizing every admissible set are likewise given by FP systems whose periodic-orbit data satisfy the structural conditions derived in that section; these constructions do not presuppose rationality. We will add a short clarifying paragraph at the beginning of §4 that explicitly separates the two parts of the argument and states that the least-period classification holds for general FP systems, including those for which the zeta-function derivative is not rational. This revision will remove any ambiguity about the scope of the claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorem

full rationale

The central claims apply the classical Skolem-Mahler-Lech theorem (an independent number-theoretic result from the 1930s-1950s) to sequences satisfying linear recurrences arising from rational logarithmic derivatives of the Artin-Mazur zeta function. The classification of least periods for finitely presented systems extends prior results on shifts of finite type via explicit constructions that realize the admissible sets. No equation or step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the argument is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the authors' own prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the Skolem-Mahler-Lech theorem applied under the rationality hypothesis and on prior results about finitely presented systems; no free parameters or new postulated entities are introduced.

axioms (1)

standard math Skolem-Mahler-Lech theorem
Invoked to classify periods n once the logarithmic derivative of the Artin-Mazur zeta function is assumed rational.

pith-pipeline@v0.9.0 · 5692 in / 1343 out tokens · 44776 ms · 2026-05-18T07:44:21.889010+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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 period n.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.