arxiv: 2605.00190 · v2 · submitted 2026-04-30 · 🧮 math.DS

Recognition: unknown

Characterisation of Stability for Interval Translation Maps

Kostiantyn Drach, Leon Staresinic, Sebastian van Strien

Pith reviewed 2026-05-09 19:52 UTC · model grok-4.3

classification 🧮 math.DS

keywords interval translation mapsstabilityabsence of critical connectionsmatchinginterval exchange transformationspiecewise translationsnon-invertible mapsdynamical systems

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The pith

Stability for general interval translation maps is characterized by the absence of critical connections together with matching.

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

The paper introduces a definition of stability that applies to interval translation maps, which are piecewise translations on an interval whose images are allowed to overlap. It proves that a map is stable under this definition precisely when it has no critical connections and satisfies the matching property. A reader would care because this equivalence supplies the first concrete tool for deciding when a non-invertible map remains robust under small perturbations, extending ideas previously limited to bijective interval exchanges. The two properties are chosen because they arise directly from the orbit structure of the map itself.

Core claim

An interval translation map is a piecewise translation defined on a finite partition of an interval, without the requirement that the images remain disjoint. We formulate an appropriate notion of stability for these general, non-bijective maps and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.

What carries the argument

The stability notion for interval translation maps, proved equivalent to the simultaneous absence of critical connections and the presence of matching.

If this is right

Stability of any concrete ITM can be checked by verifying only the two listed properties.
The same conditions supply a route to extend existing stability results from bijective interval exchanges to the larger class of ITMs.
Small perturbations of a stable ITM remain topologically conjugate to the original map.
The characterisation separates the combinatorial data of the partition from the translation lengths in a way that makes robustness testable.

Where Pith is reading between the lines

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

The same two conditions may serve as a template for defining stability in other families of piecewise maps that permit overlaps.
Numerical algorithms could be built to detect critical connections by tracking orbits of the partition endpoints.
If matching holds, the non-invertible branches still produce a well-defined itinerary structure that survives perturbation.

Load-bearing premise

That the newly introduced definition of stability is the dynamically natural and appropriate one for maps that are not required to be bijective.

What would settle it

An explicit interval translation map that satisfies absence of critical connections and matching yet fails to be stable under the paper's definition, or the converse.

Figures

Figures reproduced from arXiv: 2605.00190 by Kostiantyn Drach, Leon Staresinic, Sebastian van Strien.

**Figure 4.1.** Figure 4.1: A particular map as in Example 4.5 with β0 = 0, β1 = 1/3, β2 = 2/3, β3 = 1, γ1 = 1/3, γ2 = 1/7, γ3 = −1/2. The intervals J − and T(J −) are shown in pink and the intervals J +, T(J +), T 2 (J +) are drawn in blue. elusive, and we will need to discuss it a bit before giving the definitions of the required dynamical properties. The following example illustrates the simplest way in which the set X can becom… view at source ↗

**Figure 4.2.** Figure 4.2: A map T as in Example 4.6, with k1 = k2 = 1, β∗ = β1 and β∗∗ = β2. The perturbation increases γ1 by ϵ, decreases γ3 by ϵ and leaves the other parameters unchanged. distance from X is bounded from below by a positive constant. Therefore X(T˜) and X(T) cannot be close for a sufficiently small perturbation. Thus T is not a stable map. This example is the simplest case of a more general phenomenon called gho… view at source ↗

**Figure 4.3.** Figure 4.3: Full orbit of an interval J for which the return map RJ has four continuity intervals and satisfies A1 and A2. Lemma 4.9 (Infinite ghost trees) The ghost tree GT (β) of some discontinuity β is infinite if β appears more than once as a vertex in GT (β). This is exactly the case in Example 4.6. Proof. If β appears again as a vertex on some level n > 0, then all of the levels that appeared before the second… view at source ↗

**Figure 4.4.** Figure 4.4: Full orbit of an interval J satisfying Matching. An analogous property, also called the Matching, has been discussed in slightly different contexts (see [Bru+19], [BCK17], [NN08]). If T satisfies Matching, then the return map to every interval component J of X is either a rotation or the identity, which is a very restrictive condition. With all of the definitions out of the way, we can now recall the sta… view at source ↗

**Figure 5.1.** Figure 5.1: An example of a map for which an interval component J of X violates A1 at the point a + 2 . The dynamics of the point a + 2 and the corresponding interval are in blue. One would expect that perturbing the parameters (γ β) in such a way that β + does not land on β + ∗ would result in a hole for the return map to RJ , as in [PITH_FULL_IMAGE:figures/full_fig_p017_5_1.png] view at source ↗

**Figure 5.2.** Figure 5.2: The red arrows indicate the dynamical changes after perturbation: the orbit of a + 2 has moved to the left of β + ∗ . Assuming that the remaining dynamics of the return map stay the same, it is expected that now only the smaller interval J3 \ Jϵ returns to J, creating a hole [PITH_FULL_IMAGE:figures/full_fig_p018_5_2.png] view at source ↗

**Figure 5.3.** Figure 5.3: The perturbation could also cause the orbit of Jϵ to eventually return to J and fill the hole from [PITH_FULL_IMAGE:figures/full_fig_p018_5_3.png] view at source ↗

**Figure 5.4.** Figure 5.4: Perturbation of a return map RJ satisfying A1, A2 and Matching. The endpoints of the interval J and the point a move continuously, and the dynamical structure of RJ remains the same. a slightly smaller interval J ′ containing x, then its itinerary up to time n will not change for all sufficiently small perturbations. We would like to have the same conclusions for itineraries of critical points. For this… view at source ↗

read the original abstract

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we formulate an appropriate notion of stability for general interval translation mappings and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.

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 characterizes stability for non-bijective interval translation maps via absence of critical connections and matching.

read the letter

The paper gives a characterization of stability for interval translation maps that aren't bijective. Stability holds precisely when there are no critical connections and the matching condition is met. This extends earlier work on interval exchange transformations by handling cases where the images of the subintervals can overlap. The authors set up the definitions so that the two properties make sense for the general non-invertible maps, and they prove the equivalence using the orbit structure. What stands out is how they keep the conditions tied to the dynamics of the orbits rather than adding technical restrictions. The argument seems to use the structure of the piecewise translations directly, which keeps it clean and avoids circularity. The soft spot is small. The notion of stability they chose is presented as natural, but it would help to see a bit more on why this is the right one for applications in dynamics. Alternatives based on perturbation of the partition or the translation lengths might give different pictures, though the paper focuses on this version as the foundational step. The definitions appear well-chosen to avoid the issues that come with non-bijectivity, such as points with multiple preimages. This is for people studying one-dimensional maps and their ergodic properties. Anyone familiar with IETs will see the value in the generalization and can use it as a base for further results on non-invertible cases. The paper shows solid thinking on the topic and deserves a serious referee. I would recommend sending it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper defines interval translation maps (ITMs) as piecewise translations on a finite partition of an interval, without requiring bijectivity (in contrast to interval exchange transformations). It introduces a notion of stability for general ITMs and proves that this stability is equivalent to the conjunction of two properties: the absence of critical connections and matching. The result is positioned as a foundational step toward a stability theory for non-invertible ITMs.

Significance. If the equivalence holds, the work is significant as it supplies a dynamically natural characterization that extends stability concepts from invertible IETs to the broader class of ITMs. The self-contained definitions and proof structure provide a concrete basis for further results on orbit structure and robustness in non-bijective interval maps.

minor comments (2)

[Abstract] The abstract states the main result but does not reference the theorem number or give a one-sentence version of the precise equivalence; adding this would help readers locate the claim immediately.
[Introduction] Notation for the partition intervals I_1, …, I_r and the translation lengths is introduced without an explicit diagram or example in the opening sections; a small illustrative figure would clarify the non-bijective case for readers new to ITMs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance as a foundational step in the stability theory for non-invertible ITMs, and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring rebuttal or clarification at this stage. We are happy to incorporate any minor changes once they are specified.

Circularity Check

0 steps flagged

No significant circularity; definition plus equivalence theorem is self-contained

full rationale

The paper introduces a new definition of stability for non-bijective ITMs and proves this definition is equivalent to the conjunction of two other dynamically defined properties (Absence of Critical Connections and Matching). This is a standard characterisation theorem whose proof is developed from the internal orbit structure of ITMs; it does not reduce any claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The abstract and described structure supply all necessary definitions and lemmas without external uniqueness theorems or ansatzes imported from prior work by the same authors. A score of 0 is therefore appropriate; the only softness lies in whether the chosen stability notion is the most natural one, which is a modelling choice rather than a circularity issue.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so concrete free parameters, axioms, or invented entities cannot be extracted; the work introduces new definitions (stability, absence of critical connections, matching) that are internal to the paper rather than external postulates.

axioms (1)

standard math Standard definitions and basic properties of piecewise continuous interval maps and dynamical systems
The paper builds on classical notions from dynamical systems without stating new background axioms.

pith-pipeline@v0.9.0 · 5441 in / 1138 out tokens · 43054 ms · 2026-05-09T19:52:20.895393+00:00 · methodology

discussion (0)

Reference graph

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