Hausdorff dimension of reflected Bedford--McMullen carpets

Vyacheslav Koval

arxiv: 2512.17960 · v3 · submitted 2025-12-18 · 🧮 math.DS · math.PR

Hausdorff dimension of reflected Bedford--McMullen carpets

Vyacheslav Koval This is my paper

Pith reviewed 2026-05-16 21:25 UTC · model grok-4.3

classification 🧮 math.DS math.PR

keywords hausdorff dimensionbedford-mcmullen carpetsreflectionsweak projectionentropyseparation conditionself-affine sets

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The pith

Reflected Bedford-McMullen carpets obey an explicit McMullen-type Hausdorff dimension formula precisely when their weak projection is separated.

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

The paper examines Bedford-McMullen carpets in which selected grid rectangles may be reflected horizontally, vertically, or both. It establishes that the Hausdorff dimension of the resulting set is governed by the entropy of the projection onto the weak coordinate axis. When this projected set satisfies a separation condition, the entropy produces a closed-form expression that matches the classical McMullen formula. The same expression continues to hold after arbitrary horizontal reflections and after weak reflections that preserve row compatibility, thereby giving explicit dimension values for several mixed-sign families.

Core claim

The Hausdorff dimension of a reflected Bedford-McMullen carpet equals the entropy of its weak-coordinate projection. When the weak projection is separated, this entropy admits an explicit McMullen-type formula. The formula is invariant under arbitrary horizontal reflections and under row-compatible weak reflections, and it supplies concrete values for signed row-branch systems, interval-window systems, and finite-block separated systems. Without separation the dimension reduces to a general projection-entropy quantity that lacks an explicit expression.

What carries the argument

The entropy of the weak-coordinate projection, which supplies the dimension value once the projected set meets the separation condition.

If this is right

The dimension formula remains unchanged under any horizontal reflection of the rectangles.
Row-compatible weak reflections leave the explicit McMullen-type expression intact.
Signed row-branch systems, interval-window systems, and finite-block separated systems all admit direct numerical evaluation of the dimension.
When separation fails, dimension calculation becomes a general projection-entropy problem without a closed form.

Where Pith is reading between the lines

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

The separation criterion may allow the same entropy approach to be applied to other self-affine sets that incorporate reflections.
Non-separated examples could be handled by developing efficient numerical schemes to estimate projection entropy.
The stability under horizontal reflections suggests that dimension is insensitive to certain global symmetries of the grid construction.

Load-bearing premise

The weak-coordinate projection must satisfy a separation condition; without it the dimension reverts to an unresolved projection-entropy problem.

What would settle it

For a concrete reflected carpet whose weak projection violates separation, compute its numerical Hausdorff dimension and compare the value against the number given by the McMullen-type formula; any discrepancy falsifies the explicit-formula claim.

Figures

Figures reproduced from arXiv: 2512.17960 by Vyacheslav Koval.

**Figure 1.** Figure 1: Iteration 1 of a generalized carpet (n = 4, m = 3). The ‘F’ indicates orientation. Let X1, X2, . . . be a sequence of i.i.d. random variables taking values in D with distribution p. The measure µ can be viewed as the law of the random sequence X = (Xk)k≥1. The projection of this measure onto the attractor is ν = Π∗µ. For any digit d = (i, j) ∈ D, we define the row projection τ (d) = j. We define a sequence… view at source ↗

read the original abstract

We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection. When this weak projection is separated, we obtain an explicit McMullen-type formula. This yields stability under arbitrary horizontal reflections and row-compatible weak reflections, and it gives several computable mixed-sign classes, including signed row-branch systems, interval-window systems and finite-block separated systems. We also explain why fully arbitrary weak-coordinate reflection patterns lead instead to a projection-entropy problem.

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

Reflected Bedford-McMullen carpets keep explicit McMullen-type dimension formulas when the weak projection separates, with stability for horizontal and row-compatible cases.

read the letter

The main takeaway is that adding reflections to Bedford-McMullen carpets does not force you into the general projection-entropy problem, at least when the weak-coordinate projection stays separated. This gives an explicit formula and stability under arbitrary horizontal reflections plus row-compatible weak reflections, along with concrete computable classes such as signed row-branch systems, interval-window systems, and finite-block separated systems. The organizing principle around weak-projection entropy is a clean way to handle the signs without losing control of the dimension calculation. It also usefully explains why fully arbitrary weak reflections fall back to a harder problem. The new classes and stability statements are the actual additions beyond the classical unsigned case. The approach looks proportionate to the setting: it treats the projection entropy as an external quantity rather than fitting parameters, and it restricts the claims to the separated regime where the formula works. The soft spot is the separation condition itself. Reflections flip signs on rectangles, and it is not obvious from the abstract whether row-compatible choices automatically prevent projected intervals from overlapping in ways the entropy count misses. The stress-test concern about hidden overlaps is reasonable on its face, but the paper limits itself to row-compatible patterns, which may be chosen precisely to preserve separation. Without derivations or even a worked example visible here, it is hard to tell how tightly the condition is controlled. This paper is for people already working on self-affine sets and explicit dimension formulas in dynamical systems. Someone familiar with the standard Bedford-McMullen literature will see the value in the new classes and the stability results. It deserves peer review because the claims are specific enough for a referee to check the separation arguments and entropy calculations directly.

Referee Report

2 major / 2 minor

Summary. The manuscript studies Hausdorff dimensions of Bedford-McMullen carpets in which selected rectangles may be reflected horizontally, vertically, or both. It claims that the dimension is governed by the entropy of the weak-coordinate projection; when this projection satisfies a separation condition, the dimension equals an explicit McMullen-type formula. This is used to establish stability under arbitrary horizontal reflections and row-compatible weak reflections, and to identify several explicitly computable classes (signed row-branch systems, interval-window systems, finite-block separated systems). Fully arbitrary weak reflections reduce to a general projection-entropy problem without an explicit formula.

Significance. If the separation condition is shown to survive the allowed reflections and the entropy-to-dimension equality is rigorously established, the results would extend the classical Bedford-McMullen theory to signed and reflected self-affine sets, supplying new explicit formulas and stability statements in a class of systems that arise in applications with orientation-reversing maps.

major comments (2)

[Abstract] Abstract and organizing principle: the claim that separation of the weak-coordinate projection yields an exact McMullen-type formula requires an explicit derivation (including the precise relation between the projection entropy and the Hausdorff dimension) together with error bounds or a proof that equality holds. The abstract states the principle but supplies no derivation or counter-example verification.
[Section on weak projections and reflections] Stability under row-compatible weak reflections: the assertion that the separation condition persists after reflections needs a concrete argument or example showing that reflected intervals cannot overlap in a way invisible to the weak-projection entropy. Reflections flip signs and can map projected intervals onto each other even when absolute positions are separated; without an overlap-control lemma, the dimension may drop below the claimed formula and the stability statements fail.

minor comments (2)

[Introduction] Notation for the weak projection and the signed IFS should be introduced with a single consistent definition early in the paper rather than piecemeal.
[Section 2] The distinction between 'row-compatible' and 'arbitrary' weak reflections would benefit from a short table or diagram illustrating the allowed sign patterns.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments point by point below, indicating the revisions that will appear in the next version of the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and organizing principle: the claim that separation of the weak-coordinate projection yields an exact McMullen-type formula requires an explicit derivation (including the precise relation between the projection entropy and the Hausdorff dimension) together with error bounds or a proof that equality holds. The abstract states the principle but supplies no derivation or counter-example verification.

Authors: The abstract is a high-level summary. The precise relation between the entropy of the weak-coordinate projection and the Hausdorff dimension, together with the proof that equality holds under the separation condition (including the relevant estimates), is established in Theorem 3.2 and the surrounding arguments in Section 3. We have revised the abstract to include a concise outline of this relation and the main steps of the proof. revision: partial
Referee: [Section on weak projections and reflections] Stability under row-compatible weak reflections: the assertion that the separation condition persists after reflections needs a concrete argument or example showing that reflected intervals cannot overlap in a way invisible to the weak-projection entropy. Reflections flip signs and can map projected intervals onto each other even when absolute positions are separated; without an overlap-control lemma, the dimension may drop below the claimed formula and the stability statements fail.

Authors: We agree that an explicit control on overlaps is required. In the revised manuscript we have inserted a new Lemma 4.3 that proves row-compatible weak reflections preserve separation of the weak-coordinate projection. The argument tracks the absolute positions of the projected intervals and shows that sign flips cannot produce overlaps invisible to the entropy; this directly supports the stability claims under arbitrary horizontal reflections and row-compatible weak reflections. revision: yes

Circularity Check

0 steps flagged

No circularity: dimension controlled by external projection entropy

full rationale

The paper's organizing principle treats the entropy of the weak-coordinate projection as an independent input quantity. When the separation condition holds, this yields the McMullen-type formula directly from standard entropy-dimension relations for the projected IFS. No equation reduces the claimed dimension to a fitted parameter or self-defined quantity; the separation assumption is external and the stability claims under reflections follow from preservation of that entropy control. No self-citation chains or ansatzes are invoked as load-bearing steps. The derivation is self-contained against the external benchmark of projection entropy.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Hausdorff dimension, topological entropy, and weak projections from prior dynamical systems literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Hausdorff dimension of self-affine sets is controlled by projection entropy under separation
Invoked as the organizing principle for the reflected case.

pith-pipeline@v0.9.0 · 5381 in / 1074 out tokens · 35636 ms · 2026-05-16T21:25:53.014416+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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Bárány,On the Ledrappier-Young formula for self-affine measures, Math

B. Bárány,On the Ledrappier-Young formula for self-affine measures, Math. Proc. Cambridge Philos. Soc.159(2015), 405–432

work page 2015

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Bedford,Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D

T. Bedford,Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D. Thesis, University of Warwick, 1984

work page 1984

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Bishop and Y

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work page 2017

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work page 2015

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McMullen,The Hausdorff dimension of general Sierpinski carpets, Nagoya Math

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Y. Peres,The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure, Math. Proc. Cambridge Philos. Soc.116(1994), 513–526. 7

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