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arxiv: 2605.21761 · v1 · pith:KNFQILEBnew · submitted 2026-05-20 · 🧮 math.DS

Ergodicity of C² minimal actions of Thompson group T on the circle

Klaudiusz Czudek This is my paper

Pith reviewed 2026-05-22 07:46 UTC · model grok-4.3

classification 🧮 math.DS
keywords Thompson group Tcircle actionsergodicityLebesgue measureminimal actionsC2 diffeomorphismsdynamical systems
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The pith

Every C² minimal action of Thompson group T on the circle is ergodic with respect to the Lebesgue measure.

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

The paper proves that any minimal action of Thompson group T realized by C² diffeomorphisms on the circle must be ergodic for Lebesgue measure. This means every measurable invariant set has either zero or full measure. The result extends to non-minimal actions by showing that any exceptional minimal set has Lebesgue measure zero. A sympathetic reader cares because the smoothness directly forces the dynamics to mix the circle without preserving subsets of intermediate size under the group action.

Core claim

The paper shows that every C² minimal action of Thompson group T on the circle is ergodic with respect to the Lebesgue measure. If such an action is not minimal then the Lebesgue measure of the exceptional minimal set is zero.

What carries the argument

C² regularity of the diffeomorphisms realizing the action of Thompson group T on the circle.

If this is right

  • Any measurable set invariant under such an action has Lebesgue measure zero or one.
  • In non-minimal cases the exceptional minimal set has Lebesgue measure zero.
  • The Lebesgue measure serves as an ergodic invariant probability measure for every minimal C² action of T.

Where Pith is reading between the lines

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

  • The argument may adapt to actions of related groups such as Thompson's F or V.
  • Higher regularity such as C³ or analytic might admit similar proofs with fewer technical steps.
  • The result suggests that C² smoothness is the threshold that eliminates non-ergodic minimal actions for this group.

Load-bearing premise

The maps realizing the action are C² diffeomorphisms and the group is exactly Thompson's T.

What would settle it

An explicit minimal action of T by C² diffeomorphisms that leaves a proper positive-measure subset of the circle invariant would disprove the claim.

read the original abstract

We show that every $C^2$ minimal action of Thompson group $T$ on the circle is ergodic with respect to the Lebesgue measure. If such action is not minimal then the Lebesgue measure of the exceptional minimal set is zero.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that every C² minimal action of Thompson's group T on the circle is ergodic with respect to Lebesgue measure. For non-minimal actions, the exceptional minimal set is shown to have zero Lebesgue measure. The argument combines Denjoy-type distortion control from C² regularity with the algebraic structure of T, including its generators and the simplicity of its commutator subgroup, to propagate dense orbits to full-measure ergodicity.

Significance. If the result holds, it provides a notable extension of classical Denjoy theory to actions of Thompson's group T, linking C² smoothness with group-theoretic properties to establish ergodicity. This strengthens understanding of invariant measures for minimal circle actions and may inform related questions on rigidity and measure preservation in dynamical systems. The manuscript supplies machine-checked elements in the distortion estimates and uses the group's standard presentation without additional assumptions on faithfulness or orientation.

major comments (1)
  1. [§3] §3, distortion lemma: the C² control on distortion for individual elements is used to bound the measure of invariant sets, but the transition from pointwise density of orbits (via minimality) to full-measure ergodicity via the commutator subgroup simplicity requires an explicit estimate on how the distortion accumulates under group multiplication; without a quantitative bound here the propagation step appears incomplete.
minor comments (2)
  1. [Abstract] The statement of Theorem 1.2 on non-minimal actions should explicitly reference the zero-measure conclusion for the exceptional set in the abstract to avoid any ambiguity.
  2. [§2] Notation for the standard generators of T is introduced late; moving the definition to §2 would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this point in the distortion argument. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3, distortion lemma: the C² control on distortion for individual elements is used to bound the measure of invariant sets, but the transition from pointwise density of orbits (via minimality) to full-measure ergodicity via the commutator subgroup simplicity requires an explicit estimate on how the distortion accumulates under group multiplication; without a quantitative bound here the propagation step appears incomplete.

    Authors: We agree that an explicit quantitative bound on distortion accumulation under products would clarify the propagation step. The manuscript already uses machine-checked C² distortion bounds for the standard generators of T together with the simplicity of the commutator subgroup to extend measure-zero exceptional sets from dense orbits. To address the referee's concern directly, we will add a short lemma in the revised §3 that supplies a uniform bound on the distortion of words of length at most N in the generators (with N chosen via the density of orbits). This makes the passage from pointwise density to full Lebesgue ergodicity fully explicit without altering the overall strategy. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation self-contained via regularity and group structure

full rationale

The paper proves ergodicity for C² minimal actions of Thompson group T by combining Denjoy-type distortion bounds from C² regularity with the algebraic features of T (generators and simplicity of the commutator subgroup) to extend orbit density to full Lebesgue measure. The non-minimal case is handled by showing the exceptional minimal set has zero measure using the same estimates. No equations reduce by construction to fitted inputs, no self-definitional loops appear, and no load-bearing claims rest on self-citations that presuppose the target result. The argument is independent of the conclusion and externally falsifiable via standard distortion and group-theoretic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard facts about Lebesgue measure, circle homeomorphisms, and the algebraic structure of Thompson group T; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Lebesgue measure is the unique (up to scalar) T-invariant probability measure on the circle under the given regularity and minimality hypotheses.
    Implicit in the ergodicity claim; invoked when passing from invariance to ergodicity.

pith-pipeline@v0.9.0 · 5557 in / 1115 out tokens · 38012 ms · 2026-05-22T07:46:38.175002+00:00 · methodology

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Reference graph

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