arxiv: 2605.04121 · v1 · submitted 2026-05-05 · 🧮 math.GR · math-ph· math.DS· math.GN· math.MP

Recognition: 3 theorem links

· Lean Theorem

A dynamical approach to Schur's Theorem

Francesco G. Russo, Ilaria Svampa, Sonia L'Innocente

Pith reviewed 2026-05-08 18:17 UTC · model grok-4.3

classification 🧮 math.GR math-phmath.DSmath.GNmath.MP

keywords Schur's theoremtopological entropyZ-groupsderived subgroupmaximal almost periodic groupscontinuous endomorphismstopological groups

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The pith

A Z-group with continuous endomorphisms of finite topological entropy has its closed derived subgroup with continuous endomorphisms of finite topological entropy.

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

The paper establishes a dynamical version of Schur's 1904 theorem for topological Hausdorff groups. It shows that in Z-groups, defined as maximal almost periodic groups whose closed derived subgroup is compact, the finiteness of topological entropy for continuous endomorphisms passes from the group to its closed derived subgroup. This uses the Adler-Konheim-McAndrew definition of topological entropy to reinterpret the classical algebraic result that a finite central quotient implies a finite derived subgroup. A sympathetic reader would care because the result links algebraic structure in groups to a dynamical invariant that quantifies complexity of iterations, opening a way to study these properties in continuous settings beyond discrete groups.

Core claim

We give a new dynamical interpretation of Schur's Theorem by showing that a Z-group G with continuous endomorphisms of finite topological entropy must have its closed derived subgroup with continuous endomorphisms of finite topological entropy. The argument first studies entropy properties for endomorphisms of maximal almost periodic groups with compact derived subgroup, then applies this to obtain the stated inheritance for the commutator subgroup, and finally supplies constructions and examples to show that the result generalizes the original discrete case.

What carries the argument

Topological entropy of continuous endomorphisms, applied to Z-groups (maximal almost periodic groups with compact closed derived subgroup), which carries the dynamical complexity from the full group to the derived subgroup.

If this is right

The classical Schur theorem is recovered as a special case when the groups are discrete.
The inheritance holds for all continuous endomorphisms of the group and passes directly to the closed derived subgroup.
The result applies uniformly to the class of Z-groups studied by Grosser and Moskowitz.
Constructions and examples confirm that the dynamical property behaves as a generalization of the algebraic one.

Where Pith is reading between the lines

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

The same inheritance might hold for other dynamical invariants such as topological pressure or mean dimension on the same class of groups.
One could check the claim by explicit computation of entropy on concrete examples such as compact abelian groups or matrix groups over local fields.
The approach suggests studying whether finite entropy endomorphisms preserve other subgroup properties like the center or the Frattini subgroup in Z-groups.

Load-bearing premise

The groups must be Z-groups with compact closed derived subgroup for which topological entropy of continuous endomorphisms is well-defined.

What would settle it

A single Z-group G together with one continuous endomorphism whose topological entropy is finite on G but infinite on the closed derived subgroup.

read the original abstract

A classical result of Schur of 1904 shows that an infinite (discrete) group $E$ with finite central quotient $E/Z(E)$ should have finite derived subgroup $[E,E]$. Schur's Theorem has many important consequences, which have been extensively investigated in the literature. Here we focus on topological Hausdorff groups, which are not necessarily discrete groups, and show a dynamical version of Schur's Theorem via the notion of topological entropy of Adler, Konheim and McAndrew. Their perspective follows some original intuitions of Kolmogov and Sinai from the area of the dynamical systems. Firstly, we investigate the topological entropy of continuous endomorphisms of maximal almost periodic groups whose closed derived subgroup is compact. The properties of these groups were known to Takahashi in 1952 and among them we find the $\mathsf{Z}$-groups of Grosser and Moskowitz. Secondly, we give a new dynamical interpretation of the Schur's Theorem, showing that a $\mathsf{Z}$-group $G$ with continuous endomorphisms of finite topological entropy should have closed derived subgroup $\overline{[G,G]}$ with continuous endomorphisms of finite topological entropy. Finally, we illustrate a series of constructions and examples, which allow us to justify our interpretation of Schur's Theorem as generalization of the original version.

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 a dynamical version of Schur's theorem for Z-groups using topological entropy on endomorphisms, but the extension beyond compact spaces is the part that needs checking.

read the letter

This paper reinterprets Schur's 1904 theorem in dynamical terms: a Z-group G whose continuous endomorphisms have finite Adler-Konheim-McAndrew topological entropy must have the same property for its closed derived subgroup. They work with maximal almost periodic groups whose derived subgroup is compact, the setting introduced by Grosser and Moskowitz and studied earlier by Takahashi. The claim is that finite entropy passes from the group to the derived subgroup, and they supply constructions and examples to make the statement concrete and to recover the classical discrete case as a special instance.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a dynamical analogue of Schur's 1904 theorem for topological Hausdorff groups, focusing on Z-groups (maximal almost periodic groups whose closed derived subgroup is compact). It claims that if a Z-group G admits continuous endomorphisms of finite topological entropy (in the Adler-Konheim-McAndrew sense), then the closed derived subgroup also admits continuous endomorphisms of finite topological entropy. The work first studies entropy properties for endomorphisms of such groups, then gives the dynamical reinterpretation of Schur's theorem, and finally supplies constructions and examples to support the interpretation.

Significance. If the extension of topological entropy to non-compact Z-groups is rigorously justified and the transfer of finiteness to the compact derived subgroup is established without auxiliary parameters, the result would supply a genuine dynamical generalization of Schur's theorem. The provision of explicit examples and the grounding in known properties of Z-groups (Takahashi, Grosser-Moskowitz) are positive features that could make the interpretation useful for further work at the interface of topological groups and dynamics.

major comments (2)

[Section on topological entropy for endomorphisms of MAP groups with compact derived subgroup] The definition of topological entropy for continuous endomorphisms of non-compact Z-groups must be stated explicitly (presumably in the section following the recall of the Adler-Konheim-McAndrew definition). The standard definition requires compactness; any extension via compact subsets, uniform structures, or compactifications must be shown to be independent of choices, to coincide with the classical definition on compact subgroups, and to interact correctly with the group operation so that finiteness on G implies finiteness on the compact derived subgroup. This definition is load-bearing for the central claim.
[Section giving the new dynamical interpretation of Schur's theorem] In the proof of the main dynamical Schur statement (the theorem asserting that finite-entropy endomorphisms on G imply the same for the closed derived subgroup), the argument must verify that the entropy value on the derived subgroup is controlled by the entropy on G without introducing new parameters or relying on the compactness assumption in a circular way. If the proof uses the fact that the derived subgroup is compact, this step should be isolated and checked against the chosen entropy definition.

minor comments (2)

[Abstract] The abstract refers to 'a series of constructions and examples' without indicating their scope; a single sentence summarizing the type of examples (e.g., specific Lie groups or profinite groups) would improve readability.
[Notation and preliminaries] Notation for the center Z(G) and the derived subgroup [G,G] should be checked for consistency with the Z-group terminology throughout the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. These points help clarify the presentation of the entropy extension and the structure of the main proof. We address each comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section on topological entropy for endomorphisms of MAP groups with compact derived subgroup] The definition of topological entropy for continuous endomorphisms of non-compact Z-groups must be stated explicitly (presumably in the section following the recall of the Adler-Konheim-McAndrew definition). The standard definition requires compactness; any extension via compact subsets, uniform structures, or compactifications must be shown to be independent of choices, to coincide with the classical definition on compact subgroups, and to interact correctly with the group operation so that finiteness on G implies finiteness on the compact derived subgroup. This definition is load-bearing for the central claim.

Authors: We agree that the extension of the Adler-Konheim-McAndrew definition to non-compact Z-groups requires an explicit, self-contained statement together with the necessary independence and compatibility verifications. In the revised manuscript we will insert a dedicated subsection immediately after the recall of the classical definition. There we define the entropy of a continuous endomorphism of a Z-group via the uniform structure coming from the maximal almost periodic topology, using the compactness of the derived subgroup to reduce to compact subsets. We will prove independence of the choice of exhausting compact sets by appealing to the Takahashi characterization of Z-groups, show that the definition restricts to the classical one on compact subgroups, and verify that it is compatible with the group operation and continuous homomorphisms. These steps ensure that finite entropy on G directly implies finite entropy on the closed derived subgroup without auxiliary parameters. revision: yes
Referee: [Section giving the new dynamical interpretation of Schur's theorem] In the proof of the main dynamical Schur statement (the theorem asserting that finite-entropy endomorphisms on G imply the same for the closed derived subgroup), the argument must verify that the entropy value on the derived subgroup is controlled by the entropy on G without introducing new parameters or relying on the compactness assumption in a circular way. If the proof uses the fact that the derived subgroup is compact, this step should be isolated and checked against the chosen entropy definition.

Authors: We accept the referee's request to make the logical structure of the proof fully transparent. In the revised version we will reorganize the proof of the main theorem into three clearly separated steps: (1) recall the entropy definition established in the preceding subsection, (2) isolate the single place where compactness of the closed derived subgroup is invoked (namely, to guarantee that the restriction of the endomorphism remains continuous and that the uniform structure restricts appropriately), and (3) derive the entropy bound on the derived subgroup directly from the definition on G by using the continuity of the endomorphism and the fact that the derived subgroup is a closed normal subgroup. No new parameters are introduced; the control follows from the subadditivity properties already verified for the extended definition. A short remark will be added to emphasize that the compactness step is used only after the definition has been shown to be well-defined and compatible, thereby avoiding any circularity. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained; no circular reductions to inputs or self-citations

full rationale

The paper applies the standard Adler-Konheim-McAndrew topological entropy to continuous endomorphisms of Z-groups (maximal almost periodic groups whose closed derived subgroup is compact, per Takahashi 1952 and Grosser-Moskowitz). It then offers a dynamical reading of Schur's theorem stating that finite entropy on G transfers to the closed derived subgroup. No equations redefine entropy in terms of the target conclusion, no parameters are fitted to a subset and then relabeled as prediction, and no load-bearing premise rests on a self-citation whose validity is internal to the authors' prior work. The cited group properties and entropy definition are external and independent; the result is an interpretation rather than a tautology by construction. The derivation chain therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of topological groups, maximal almost periodic groups, Z-groups, and the Adler-Konheim-McAndrew topological entropy; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Hausdorff topological groups with continuous endomorphisms
Stated as the setting for the dynamical version.
standard math Topological entropy defined for continuous endomorphisms of groups
Referenced via Adler, Konheim and McAndrew.

pith-pipeline@v0.9.0 · 5542 in / 1257 out tokens · 43407 ms · 2026-05-08T18:17:19.368381+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.Cost.FunctionalEquation (J(x)=½(x+x⁻¹)−1) washburn_uniqueness_aczel unclear
the topological entropy was adapted by Adler, Konheim, and McAndrew [2] to continuous self-maps on compact topological spaces ... H_top(φ, V) = lim sup_{n→∞} −log μ(C_n(φ, V))/n

Reference graph

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