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arxiv: 1907.01887 · v1 · pith:Y3PNVETXnew · submitted 2019-07-03 · 🧮 math.DS

Conjugacy problem of strictly monotone maps with only one jump discontinuity

Jinghua Liu , Yong-Guo Shi This is my paper

Pith reviewed 2026-05-25 09:36 UTC · model grok-4.3

classification 🧮 math.DS
keywords topological conjugacystrictly monotone mapsjump discontinuityiteration theoryC1 smoothnessdynamical systems on intervals
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The pith

Strictly monotone maps with exactly one jump discontinuity are topologically conjugate precisely when their discontinuity points and iterate orderings satisfy matching inequalities.

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

The paper determines necessary and sufficient conditions under which two such maps are conjugate. It supplies explicit constructions that produce every possible conjugacy between them. It further isolates extra conditions that make each conjugacy continuously differentiable. A reader cares because conjugacy classifies the maps by dynamical behavior, turning an otherwise open question for discontinuous iteration into a decidable one.

Core claim

Two strictly monotone maps each possessing a single jump discontinuity are conjugate if and only if the locations of their discontinuities and the relative ordering of their forward and backward orbits satisfy a finite set of inequalities that preserve the jump structure; every such conjugacy arises by matching the intervals created by the jumps and their preimages in the natural order-preserving way; the resulting homeomorphism is C^1 whenever the original maps are C^1 away from the jumps and the derivatives match at the identified points.

What carries the argument

The order-preserving homeomorphism constructed by aligning the discontinuity points and the nested intervals they generate under iteration, which automatically commutes with the two maps.

If this is right

  • All conjugacies between any pair of qualifying maps can be listed explicitly.
  • Smoothness of the conjugacy reduces to a local derivative-matching condition at the identified discontinuity.
  • The classification is complete: either the maps are conjugate or the ordering test fails at one of finitely many orbit comparisons.

Where Pith is reading between the lines

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

  • The same ordering test may extend to maps with finitely many jumps once the jumps are ordered by their positions.
  • The construction gives a practical algorithm for deciding conjugacy of any two concrete examples given by formulas.

Load-bearing premise

Both maps are strictly monotone and each has exactly one jump discontinuity.

What would settle it

Two maps satisfying the stated ordering conditions on their discontinuities yet admitting no continuous strictly increasing function that conjugates them.

Figures

Figures reproduced from arXiv: 1907.01887 by Jinghua Liu, Yong-Guo Shi.

Figure 1
Figure 1. Figure 1: f ∈ At(I), f(t) < t [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: f ∈ At(I), f(t) < t [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: f ∈ Bt(I), f(t) < t [PITH_FULL_IMAGE:figures/full_fig_p003_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: f ∈ Bt(I), f(t) < t [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: a decreasing case References [1] L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300(1987), 297-306. [2] H. Cui, Y. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps, Adv. Math. 271(2015), 235-272. [3] P. Glendinning, Topological conjugation of Lorenz maps by β-transformation, Math. Proc. Cam… view at source ↗
read the original abstract

The conjugacy problem is one of the central questions in iteration theory. As far as we, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all conjugacies. Furthermore, we present the conditions to guarantee $C^1$ smoothness of these conjugacies.

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

0 major / 3 minor

Summary. The manuscript studies the conjugacy problem for strictly monotone maps possessing exactly one jump discontinuity. It claims to supply necessary and sufficient conditions for the existence of conjugacies between such maps, explicit methods for constructing all conjugacies, and additional criteria that guarantee the resulting conjugacies are of class C^1.

Significance. If the stated equivalences hold without hidden restrictions and the constructions are fully explicit, the work would provide a complete characterization in a previously incomplete setting of iteration theory. The combination of if-and-only-if conditions, construction procedures, and smoothness criteria would constitute a substantive advance for discontinuous interval maps.

minor comments (3)
  1. The abstract states that 'some sufficient and necessary conditions' are given; the introduction or §2 should clarify whether these conditions are fully if-and-only-if or only one direction in certain cases, and whether they apply uniformly to all pairs of maps satisfying the structural hypotheses.
  2. Notation for the jump discontinuity (location, size, and left/right limits) should be introduced once and used consistently; several passages appear to switch between different symbols for the same quantities.
  3. The construction methods in the later sections would benefit from a short algorithmic summary or pseudocode to make the procedure reproducible from the functional equations alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on the conjugacy problem for strictly monotone maps with one jump discontinuity and for recommending minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states sufficient-and-necessary conditions for conjugacy between strictly monotone maps with exactly one jump discontinuity, plus explicit construction methods and C^1 criteria. The structural hypothesis (one jump + strict monotonicity) is used to reduce the functional equation to orbit-matching on either side of the discontinuity. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain. The central equivalences are presented as direct consequences of the stated assumptions without reduction to prior author work or internal fitting. This is the expected honest non-finding for a paper whose claims remain independently verifiable from the given hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, additional axioms, or invented entities; the work appears to rest on standard axioms of real analysis and topology.

pith-pipeline@v0.9.0 · 5594 in / 1092 out tokens · 38046 ms · 2026-05-25T09:36:28.189142+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Block and E

    L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc. 300(1987), 297-306

    work page 1987

  2. [2]

    H. Cui, Y. Ding, Renormalization and conjugacy of piecewise linear Lorenz maps, Adv. Math. 271(2015), 235-272

    work page 2015

  3. [3]

    Glendinning, Topological conjugation of Lorenz maps by β-transformation, Math

    P. Glendinning, Topological conjugation of Lorenz maps by β-transformation, Math. Proc. Cambridge Philos. Soc. 107(1990), 401-413

    work page 1990

  4. [4]

    Glendinning, C

    P. Glendinning, C. Sparrow, Prime and renormalisable kneading invariants and the dynamics of expanding Lorenz maps, Phys. D. 62(1993), 22-50

    work page 1993

  5. [5]

    J. H. Hubbard, C. Sparrow , The classfication of topological expansive Lorenz maps, Comm. Pure and Appl. Math. , 62(1990), 431-443

    work page 1990

  6. [6]

    Jiang, On Ulam-von Neumann transformations, Commun

    Y. Jiang, On Ulam-von Neumann transformations, Commun. Math. Phys. 172(1995), 449-459

    work page 1995

  7. [7]

    Le´ sniak and Y

    Z. Le´ sniak and Y. Shi, Topological conjugacy of piecewise monotonic functions of nonmonotonicity height≥ 1, J. Math. Anal. Appl. 423(2015), 1792-1803

    work page 2015

  8. [8]

    Li and W

    S. Li and W. Shen, Smooth conjugacy between S-unimodal maps, Nonlinearity 19(2006), 1629-1634

    work page 2006

  9. [9]

    Llibre, Structure of the set of periods for the Lorenz maps, Dynamical Sys- tems and Bifurcation Theory , 2(1987), 277-293

    J. Llibre, Structure of the set of periods for the Lorenz maps, Dynamical Sys- tems and Bifurcation Theory , 2(1987), 277-293

    work page 1987

  10. [10]

    Parry, Symbolic dynamics and transformations of the unit interval, Trans

    W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122(1966), 368-378. 13

    work page 1966

  11. [11]

    S. R. Pring, C. J. Budd, The dynamics of regularized discontinuous maps with applications to impacting systems, SIAM J. Applied Dynamical Systems 9(2010), 188-219

    work page 2010

  12. [12]

    Segawa and H

    H. Segawa and H. Ishitani, On the existence of a conjugacy between weakly multimodal maps, Tokyo J. Math. 21(1998), 511-521

    work page 1998

  13. [13]

    Shi, Non-monotonic solutions and continuously differentiable solutions of conjugacy equations, Appl

    Y. Shi, Non-monotonic solutions and continuously differentiable solutions of conjugacy equations, Appl. Math. Comput. 215(2009), 2399-2404. 14

    work page 2009