Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

Nizar El Idrissi; Samir Kabbaj

arxiv: 2210.11412 · v2 · submitted 2022-10-20 · 🧮 math.DS · math.CO

Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

Nizar El Idrissi , Samir Kabbaj This is my paper

Pith reviewed 2026-05-24 10:49 UTC · model grok-4.3

classification 🧮 math.DS math.CO

keywords invariant setsquasi-invariant setsfunction actionsfinite subsetssupersetsnatural numbersdynamical systems

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The pith

The functions f on a set I for which every finite subset is internally-k-quasi-invariant are characterized for k in {0,1}.

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

The paper extends the notions of invariant and k-quasi-invariant sets from group actions to actions generated by a single function f on an arbitrary set I. It determines the functions f such that every finite subset of I is internally-k-quasi-invariant. It also finds the functions where every finite subset has a finite internally-k-quasi-invariant superset. Special cases are considered when I is the natural numbers and the subsets are finite intervals. This parallels the investigation by Praeger for group actions.

Core claim

The paper answers questions of the following type, where k in {0,1}: what are the functions f for which every finite subset of I is internally-k-quasi-invariant? More restrictively, if I = N, what are the functions f for which every finite interval of I is internally-k-quasi-invariant? Last, what are the functions f for which every finite subset of I admits a finite internally-k-quasi-invariant superset?

What carries the argument

Internally-k-quasi-invariant sets for the action generated by iteration of a single function f on a set I, with k=0 corresponding to invariance and k=1 to internal 1-quasi-invariance.

If this is right

The functions achieving the stated maximal properties are completely described for arbitrary I.
When I equals the natural numbers the same properties hold precisely for certain restricted classes of functions on intervals.
Every finite subset admits a finite internally-k-quasi-invariant superset precisely for the functions identified in the classification.
The results supply direct analogues of Praeger's group-action statements in the single-function setting.

Where Pith is reading between the lines

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

The same questions could be posed for k greater than 1 or for actions generated by finitely many functions.
The characterizations may be used to classify discrete dynamical systems on countable sets according to their finite invariant structure.
One could check the successor function on N against the interval classification to test consistency with the stated answers.

Load-bearing premise

The notions of invariant set and internally-1-quasi-invariant set, originally developed for group actions, extend directly and meaningfully to actions generated by a single function f on an arbitrary set I.

What would settle it

An explicit function f on N together with a finite interval that fails to be internally-1-quasi-invariant, if that f is asserted by the characterization to satisfy the property for all intervals, would settle the interval case.

read the original abstract

A relaxation of the notion of invariant set, known as $k$-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function $f$ on a set $I$. We answer several questions of the following type, where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I = \mathbb{N}$, what are the functions $f$ for which every finite interval of $I$ is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which every finite subset of $I$ admits a finite internally-$k$-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions.

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

The paper gives explicit characterizations of functions f on a set I (and on N) such that finite subsets are invariant or internally-1-quasi-invariant under iteration, or admit finite supersets with the property, extending Praeger's group-action results.

read the letter

The main takeaway is a collection of characterizations for functions f where every finite subset of the domain is internally-k-quasi-invariant for k in {0,1}, or every finite subset has a finite superset that is. They also handle the restricted case on the naturals where the subsets are intervals. This is new in moving the quasi-invariant notions from group actions to the semigroup generated by iterating one function, and the paper answers the questions by direct case analysis on orbit structure under f. It does well in setting the questions up in parallel to the earlier group work and keeping the arguments parameter-free and based on explicit dynamics like cycles and transients. The extension appears to go through cleanly with no visible internal contradictions or hidden assumptions from the group setting. The soft spots are that the results stay narrow, confined to finite-set questions in this corner of discrete dynamics, so the reach is limited even if the claims hold. The definitions need to be checked in the proofs for how they behave when f is not bijective, but nothing in the structure suggests a load-bearing flaw. This is for people already working on invariant sets under set actions or combinatorial dynamics. A reader who knows Praeger's paper would see the value in the single-map adaptation. It deserves a serious referee because the questions are well-posed, the approach is direct, and the work is formally grounded in case analysis rather than loose claims.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes functions f:I→I such that every finite subset of I is internally-k-quasi-invariant (k=0,1), the functions for which every finite subset admits a finite internally-k-quasi-invariant superset, and the special case I=ℕ where the subsets are required to be intervals. The characterizations are obtained by direct case analysis on the orbit structure of f, extending Praeger's group-action results to the single-generator setting.

Significance. If the derivations hold, the paper supplies a complete, parameter-free classification of the single-map case, clarifying the distinction between invariant and 1-quasi-invariant behavior under iteration. The explicit dynamical case analysis (rather than reduction to fitted parameters or external results) is a strength, as is the parallel treatment of the interval condition on ℕ.

minor comments (3)

[§2] §2: the definition of internally-1-quasi-invariant set for a single function f is introduced via the group-action template; an explicit one-line restatement in terms of f-orbits would improve readability for readers coming from the function-dynamics side.
[Theorem 4.2] Theorem 4.2 (the superset result): the proof sketch refers to 'the same orbit decomposition as in Lemma 3.4' but does not restate the relevant orbit types; adding a one-sentence reminder would make the argument self-contained.
[§5] The comparison table (if present) or the final summary section would benefit from a side-by-side listing of the group-action versus single-function conditions to highlight exactly where the extension changes the statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. No major comments were listed in the report, so there are no specific points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity; self-contained characterization

full rationale

The paper performs explicit case analysis and characterization of functions f on sets I (and on N) with respect to internally-k-quasi-invariant sets for k=0,1. Definitions are introduced directly for the single-function case rather than being defined in terms of the target results. The parallel to Praeger's group-action results is an external citation, not a self-citation chain or load-bearing premise. No parameters are fitted, no predictions reduce to inputs by construction, and no ansatz is smuggled via prior self-work. The derivation is parameter-free and consists of direct proofs on the dynamics of f.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard set-theoretic foundations for defining functions, subsets, and invariance; no free parameters, ad-hoc axioms, or invented entities are visible from the abstract.

axioms (1)

standard math Standard axioms of set theory (ZFC) for defining sets, functions, and subsets.
Invoked implicitly when discussing subsets of I and actions of f.

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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/AbsoluteFloorClosure reality_from_one_distinction unclear

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

what are the functions f for which every finite subset of I is internally-k-quasi-invariant? ... parallels ... Praeger in the context of group actions

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