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arxiv: 2606.30629 · v1 · pith:SL2B6IAMnew · submitted 2026-06-29 · 🧮 math.DS

Rauzy-Veech Induction for Infinite-Type IETs

Charles Fougeron , Sophie Schmidhuber This is my paper

Pith reviewed 2026-06-30 03:01 UTC · model grok-4.3

classification 🧮 math.DS
keywords interval exchange transformationsunique ergodicityRauzy-Veech inductioninfinite-type surfacestail-reversing permutationsminimal interval exchangesinfinite genus translation surfaces
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The pith

For any tail-reversing permutation, the lengths making an infinite-type IET uniquely ergodic contain a dense Gδ set in the simplex under the ℓ¹ topology.

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

The paper generalizes Rauzy-Veech induction from finite to a broad class of infinite-type interval exchange transformations that arise from finite-area infinite-genus translation surfaces. It establishes a minimality criterion that extends Keane's criterion and then specializes to tail-reversing permutations. Through a combinatorial study of the associated infinite Rauzy diagrams it proves that unique ergodicity is topologically generic in the space of length vectors. An explicit sufficient condition for unique ergodicity is also extracted from the diameters of the induction matrices.

Core claim

We generalize Rauzy-Veech induction to a large class of infinite-type IETs, prove a minimality criterion that extends Keane's criterion, and show that for every tail-reversing permutation the set of length vectors in the simplex Δ for which the corresponding IET is uniquely ergodic contains a dense Gδ set with respect to the ℓ¹ topology. The result is obtained by combinatorial analysis of the infinite-type Rauzy diagrams; an explicit condition on the diameter of the induction matrices is derived as a sufficient criterion for unique ergodicity.

What carries the argument

Infinite-type Rauzy diagrams together with their induction matrices, specialized to tail-reversing permutations.

If this is right

  • A minimality criterion holds for the generalized class of infinite-type IETs.
  • Unique ergodicity is topologically generic among length assignments for every fixed tail-reversing permutation.
  • An explicit numerical test on the diameters of successive induction matrices decides unique ergodicity for tail-reversing cases.
  • The same combinatorial machinery applies directly to concrete examples such as the Baker surface.

Where Pith is reading between the lines

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

  • The genericity result suggests that typical finite-area infinite-genus translation surfaces may exhibit unique ergodicity for their horizontal or vertical flows.
  • The method supplies a template that could be adapted to other infinite combinatorial types once their Rauzy diagrams are classified.
  • Diameter bounds on induction matrices may yield effective Diophantine conditions usable in numerical experiments on concrete infinite IETs.

Load-bearing premise

The generalized Rauzy-Veech induction procedure and the combinatorial description of the infinite Rauzy diagrams accurately reflect the orbit dynamics of tail-reversing infinite-type IETs.

What would settle it

An explicit length vector in the simplex for some tail-reversing permutation such that the induced map fails to be minimal or uniquely ergodic while the Rauzy diagram analysis predicts it should belong to the dense Gδ set.

Figures

Figures reproduced from arXiv: 2606.30629 by Charles Fougeron, Sophie Schmidhuber.

Figure 1
Figure 1. Figure 1: The inverse permutation. 1 arXiv:2606.30629v1 [math.DS] 29 Jun 2026 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: An infinite-type IET without saddle connections and an in￾variant Cantor set. From "Infinite translation surfaces in the wild" by Delecroix, Hubert and Valdez [DHV24], p. 215, reprinted with permis￾sion. 2.3.3. Failure of Keane’s criterion and Maier’s Theorem in the infinite setting. The pre￾vious construction yields an example of an infinite-type IET T which has no saddle connections but which is also not… view at source ↗
Figure 4
Figure 4. Figure 4: Case 1a) where I t At is a tail interval and wins on top. We reveal its outermost letter α1. b) (single interval wins) if I t At is not a tail interval, we define T1 to be the first return map of T to [0, inf(I b Ab )] (which is explicitly given in Definition 3.1(i) where w = At and L = Ab). We change the bottom permutation πb,1 as described in Definition 3.1(i) and keep all other combinatorial data the sa… view at source ↗
Figure 5
Figure 5. Figure 5: Case 1b) where I t At is not a tail interval and wins on top. We restrict T to [0, inf(I b Ab )]. 2. Bottom wins. If inf(I t At ) > inf(I b Ab ), we say that bottom wins. Then ◦ (tail interval wins) if I b Ab is a tail interval, we set T1 = T and set Π1 to be the permutation obtained from Π by revealing the outermost letter of Ab, as well as setting λ1 = λ. ◦ (single interval wins) if I b Ab is not a tail … view at source ↗
read the original abstract

We consider infinite-type IETs arising from elementary examples of finite-area translation surfaces of infinite genus such as the Baker's surface. We call such IETs tail-reversing and we show that for any tail-reversing permutation the subset of the simplex of lengths $\Delta$ for which the corresponding infinite-type IET is uniquely ergodic contains a dense $G_{\delta}$ set with respect to the $\ell^1$-topology. To this end, we generalize Rauzy-Veech induction to a large class of infinite-type IETs, where we prove a minimality criterion as a generalization of Keane's criterion in the finite setting. We then restrict ourselves to tail-reversing IETs and obtain our genericity result through a combinatorial analysis of their infinite-type Rauzy diagrams. Moreover, we derive an explicit condition for a tail-reversing IET to be uniquely ergodic by studying the diameter of its induction matrices.

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

2 major / 2 minor

Summary. The paper generalizes Rauzy-Veech induction to a class of infinite-type interval exchange transformations (IETs) arising from finite-area infinite-genus translation surfaces (e.g., the Baker surface). These are termed tail-reversing IETs. The authors prove a minimality criterion that extends Keane's criterion, then use combinatorial analysis of the associated infinite Rauzy diagrams to show that, for every tail-reversing permutation, the set of length vectors in the simplex Δ for which the IET is uniquely ergodic contains a dense Gδ subset in the ℓ¹ topology. They also derive an explicit unique-ergodicity criterion in terms of the diameters of the induction matrices.

Significance. If the technical claims hold, the work supplies the first genericity result for unique ergodicity of infinite-type IETs and introduces a combinatorial framework for infinite Rauzy diagrams. This extends the classical Keane–Masur–Veech theory to infinite-genus surfaces and provides an explicit diameter-based criterion that may be checkable in concrete examples.

major comments (2)
  1. [§4] §4, the statement of the minimality criterion: the proof that an infinite Rauzy diagram being connected and having no periodic orbits implies minimality for all length vectors in a full-measure set appears to rely on an infinite-product argument for the return times; it is not immediate that the same argument controls the infinite-type case without a uniform bound on the number of intervals visited per step.
  2. [§6] §6, the diameter estimates used for the Gδ argument: the claim that the product of induction matrices has diameter tending to zero for a dense Gδ set of lengths is central to the genericity result, yet the argument invokes a countable intersection over all finite sub-diagrams; it is unclear whether the ℓ¹ topology ensures that the exceptional set remains meager when the diagram is infinite.
minor comments (2)
  1. [Introduction] Notation for the infinite simplex Δ and the ℓ¹ topology should be introduced with an explicit definition of the metric before the statement of the main theorem.
  2. [§2] The definition of 'tail-reversing' permutation is given only in the abstract and §2; a self-contained paragraph with an example (e.g., the Baker surface permutation) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these points in the proofs. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4] §4, the statement of the minimality criterion: the proof that an infinite Rauzy diagram being connected and having no periodic orbits implies minimality for all length vectors in a full-measure set appears to rely on an infinite-product argument for the return times; it is not immediate that the same argument controls the infinite-type case without a uniform bound on the number of intervals visited per step.

    Authors: The tail-reversing combinatorial structure ensures that induction steps affect only finitely many intervals before the reversing tail takes over, yielding a uniform bound on intervals visited per step that is independent of the length vector. This bound is implicit in the definition of tail-reversing permutations and allows the infinite-product argument to carry over directly. We will insert an explicit lemma in the revised §4 stating this bound and verifying the product convergence. revision: partial

  2. Referee: [§6] §6, the diameter estimates used for the Gδ argument: the claim that the product of induction matrices has diameter tending to zero for a dense Gδ set of lengths is central to the genericity result, yet the argument invokes a countable intersection over all finite sub-diagrams; it is unclear whether the ℓ¹ topology ensures that the exceptional set remains meager when the diagram is infinite.

    Authors: The ℓ¹ simplex is a complete metric space and therefore Baire. Each finite sub-diagram condition defines a dense open set in the ℓ¹ topology because any length vector can be approximated in ℓ¹ by one whose finite initial segment satisfies the finite-type unique-ergodicity criterion while the tail remains positive. The countable intersection over all finite sub-diagrams is therefore dense Gδ. We will add a short lemma in the revised §6 confirming that the exceptional sets remain meager under the ℓ¹ metric. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes its genericity result via a generalization of Rauzy-Veech induction to infinite-type IETs, a minimality criterion extending Keane's criterion, and combinatorial analysis of infinite Rauzy diagrams for tail-reversing permutations, followed by diameter estimates on induction matrices. No quoted steps in the abstract or description reduce any prediction or central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The combinatorial and topological arguments are presented as independent of the target result, yielding a dense Gδ set under the ℓ¹ topology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are introduced in the abstract; the work relies on standard background results from finite-type IET theory.

axioms (1)
  • standard math Standard properties of interval exchange transformations, translation surfaces, and Rauzy-Veech induction in the finite case
    The generalization and minimality criterion build directly on these background facts.

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Reference graph

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