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arxiv: 1906.09804 · v1 · pith:JHYT7UBDnew · submitted 2019-06-24 · 🧮 math.DS

A Besicovitch-Morse function preserving the Lebesgue measure

Jozef Bobok , Serge Troubetzkoy (I2M) This is my paper

Pith reviewed 2026-05-25 17:08 UTC · model grok-4.3

classification 🧮 math.DS
keywords Besicovitch-Morse functionLebesgue measuremeasure preservationfirst categorycontinuous functionsergodic theorydynamical systems
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The pith

A Besicovitch-Morse function that preserves Lebesgue measure exists, though such functions form a meager set among continuous measure-preserving maps.

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

The paper constructs a Besicovitch-Morse function that preserves Lebesgue measure, continuing the study of non-differentiable maps in ergodic theory. It demonstrates that this irregular map respects the measure while being non-differentiable. Additionally, it proves that Besicovitch functions are of first category, meaning they are rare in the topological sense, within the space of all continuous Lebesgue measure-preserving functions. This indicates that measure preservation is compatible with certain types of non-differentiability but does not hold generically for these specific functions.

Core claim

We construct a Besicovitch-Morse function map which preserves the Lebesgue measure. We also show that the set of Besicovitch functions is of first category in the set of continuous functions which preserve the Lebesgue measure.

What carries the argument

The Besicovitch-Morse function, a non-differentiable map with specific properties from prior definitions, that is shown to preserve Lebesgue measure.

If this is right

  • Besicovitch-Morse functions can occur as measure-preserving maps in dynamical systems.
  • The set of such functions is meager in the space of continuous Lebesgue measure-preserving maps.
  • Non-differentiable maps of this type are compatible with ergodic theory frameworks.
  • These functions are exceptional rather than typical among continuous maps preserving measure.

Where Pith is reading between the lines

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

  • Similar constructions might apply to other measures or spaces beyond the interval.
  • The category result suggests that generic continuous measure-preserving maps are differentiable or of different irregularity type.
  • Extensions could involve higher-dimensional maps or different notions of non-differentiability.

Load-bearing premise

The prior definitions of Besicovitch-Morse functions and the space of continuous Lebesgue measure-preserving maps from the referenced work admit the required construction.

What would settle it

An explicit demonstration that no Besicovitch-Morse function can preserve Lebesgue measure, or that the set of Besicovitch functions is not meager in the space of continuous measure-preserving maps.

Figures

Figures reproduced from arXiv: 1906.09804 by Jozef Bobok, Serge Troubetzkoy (I2M).

Figure 1
Figure 1. Figure 1: The map f0,σ Given a map f : I → I and x, y ∈ I, x 6= y, define R(f, x, y) := f(x) − f(y) x − y . We consider a continuous nondecreasing function f0,σ : [0, 1/2] → I satisfying f0,σ(0) = 0, f0,σ(1/2) = 1, f0,σ constant on every interval rm,p and satisfying for each dm,p := [c, d], bm,p (1) R(f0,σ, bm,p, c) = R(f0,σ, d, c). Notice that this number is at least 2 for every dm,p. The function f0,σ is a Cantor … view at source ↗
Figure 2
Figure 2. Figure 2: ∆n oriented upwards (left) and downwards (right) Consider [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The upper left endpoint of the small triangle has coordinates (an+1, f(an+1)). Assume that f(an+1) > f(an) + 2αnhn; using (14) we get (15) f(an+1) − f(an) = αn+1hn+1 + αnhn > 2αnhn hence again from (14) αn+1hn+1 > αnhn > (1 − αn+1)hn+1 and αn+1 > 1/2, a contradiction with our choice of αn+1. It shows that f(an+1) 6 f(an) + 2αnhn [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: i) g (0) 1 and f0, ii) g (1) 1 (the segment r1,1 is drawn for comparison), iii) g (2) 1 (the segment r2,1 is drawn for comparison) where L ∗ n−1 denotes the set of all (n − 1)st L-segments and their counterparts in [1/2, 1]. On each element of L ∗ n−1 instead of rescaled step triangle we use a rescaled tent map of the same base, height and orientation. Then limn gn = limn fn = f, so it is sufficient to sho… view at source ↗
read the original abstract

We continue the investigation of which non-dierentiable maps can occur in the framework of ergodic theory started in [2]. We construct a Besicovitch-Morse function map which preserves the Lebesgue measure. We also show that the set of Besicovitch functions is of rst category in the set of continuous functions which preserve the Lebesgue measure.

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 / 1 minor

Summary. The manuscript constructs an explicit Besicovitch-Morse map that preserves Lebesgue measure and proves via a Baire-category argument that the set of all Besicovitch functions is meager in the space of continuous Lebesgue-measure-preserving maps.

Significance. If the construction satisfies the definition from the cited reference [2] and the category argument is correctly formulated, the result supplies a concrete example of a non-differentiable measure-preserving map and establishes that such maps are topologically exceptional among continuous measure-preserving maps. This continues the program initiated in [2] with an explicit existence statement and a genericity result.

major comments (2)
  1. [Construction (main body)] The explicit construction must be verified against every clause of the Besicovitch-Morse definition given in reference [2]; any failure to meet a required property while simultaneously preserving Lebesgue measure would invalidate the existence claim.
  2. [Category argument] The Baire-category argument requires that the ambient space of continuous Lebesgue-measure-preserving maps be a complete metric space under the chosen topology; the manuscript must specify the metric and confirm completeness so that the Baire theorem applies directly.
minor comments (1)
  1. [Abstract] Abstract contains typographical errors: 'dierentiable' should read 'differentiable' and 'rst' should read 'first'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed report and for highlighting these important points regarding the construction and the category argument. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Construction (main body)] The explicit construction must be verified against every clause of the Besicovitch-Morse definition given in reference [2]; any failure to meet a required property while simultaneously preserving Lebesgue measure would invalidate the existence claim.

    Authors: Our construction is built to satisfy the definition from [2] at each step, with measure preservation ensured by the choice of the maps used. To address the referee's concern directly, we will add an explicit verification subsection in the revised manuscript that checks each clause of the definition against the constructed map. revision: yes

  2. Referee: [Category argument] The Baire-category argument requires that the ambient space of continuous Lebesgue-measure-preserving maps be a complete metric space under the chosen topology; the manuscript must specify the metric and confirm completeness so that the Baire theorem applies directly.

    Authors: We acknowledge that the completeness of the space should be stated explicitly. The space is the set of continuous maps from the circle to itself that preserve Lebesgue measure, endowed with the uniform metric. This space is complete because it is a closed subset of the complete space of all continuous maps. We will include this specification and confirmation in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Explicit construction and category argument independent of self-citation chain

full rationale

The paper states it continues work from [2] on non-differentiable maps in ergodic theory, then gives an explicit construction of a Besicovitch-Morse map preserving Lebesgue measure plus a Baire-category argument that such maps are first category among continuous measure-preserving maps. No equation or step reduces the claimed existence or category result to a fitted parameter, self-definition, or load-bearing self-citation; the reference supplies only the ambient definitions and space, while the new content is the construction and verification. This matches the normal non-circular case for a construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract alone; no explicit free parameters, axioms, or invented entities are visible in the provided text.

pith-pipeline@v0.9.0 · 5577 in / 1093 out tokens · 23337 ms · 2026-05-25T17:08:28.137823+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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    work page 1937