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arxiv: 2606.06012 · v2 · pith:WMFVL65Snew · submitted 2026-06-04 · 🧮 math.DS

Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets

Nima Alibabaei , Masaki Tsukamoto This is my paper

Pith reviewed 2026-06-27 23:23 UTC · model grok-4.3

classification 🧮 math.DS
keywords weighted topological entropyBedford-McMullen carpetsHausdorff dimensionrandom translatesFeng-Huang entropyself-affine setsfactor mapsKenyon-Peres formula
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The pith

The Hausdorff dimension of intersections of random translates of two Bedford-McMullen carpets is given by an explicit formula extending the Kenyon-Peres result to self-affine sets.

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

The paper proves a relativised variational principle for Feng-Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with Yin's theorem, this produces an almost-everywhere equivalence between the weighted entropy and its combinatorial version on fibers. The equivalence is applied to calculate the Hausdorff dimension of the intersection of random translates of two Bedford-McMullen carpets. A sympathetic reader cares because the result supplies a computable dimension formula in a setting where self-affine geometry had previously lacked such an extension from the self-similar case.

Core claim

We establish a relativised variational principle for the Feng-Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng-Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford-McMullen carpets. The resulting formula extends the Kenyon-Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.

What carries the argument

Relativised variational principle for Feng-Huang weighted topological entropy associated with a factor map, which produces fiberwise entropy equivalence almost everywhere when paired with Yin's theorem.

If this is right

  • The dimension of the intersection is determined by the common value of the Feng-Huang entropy and its combinatorial counterpart on almost every fiber.
  • The formula applies directly to self-affine carpets rather than being restricted to self-similar sets.
  • The same entropy equivalence supplies a new problem on the growth rates of random matrix products.
  • The relativised variational principle holds for arbitrary factor maps between dynamical systems.

Where Pith is reading between the lines

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

  • The same fiberwise entropy comparison may be usable for dimension calculations involving other families of self-affine sets.
  • The random-matrix-product question raised in the paper could be studied independently via multiplicative ergodic theory.
  • The technique might adapt to intersections of more than two carpets or to translates chosen according to different probability measures.

Load-bearing premise

Yin's theorem applies to the fibers arising from the factor maps associated with the Bedford-McMullen carpet systems.

What would settle it

An explicit numerical or symbolic computation for a concrete pair of Bedford-McMullen carpets in which the actual Hausdorff dimension of a typical random-translate intersection differs from the value given by the entropy formula.

Figures

Figures reproduced from arXiv: 2606.06012 by Masaki Tsukamoto, Nima Alibabaei.

Figure 1
Figure 1. Figure 1: The Bedford–McMullen carpet for a = 3, b = 2 and D = {(0, 0),(1, 1),(2, 0)}. dimension is given by dimH X = log2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Bedford–McMullen carpet for a = 3, b = 2 and the digit set {(0, 0),(1, 0),(2, 0),(1, 1)}. See [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

We establish a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a recent theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang entropy and its combinatorial version on fibers. As an application, we compute the Hausdorff dimension of the intersection of random translates of two Bedford--McMullen carpets. The resulting formula extends the Kenyon--Peres formula from the self-similar to the self-affine setting, and also points to a new problem concerning random matrix products.

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 paper establishes a relativised variational principle for Feng-Huang weighted topological entropy under a factor map between dynamical systems. Combined with a theorem of Yin, this yields an a.e. equivalence between the weighted entropy and its combinatorial version on fibers. As an application, the authors compute the Hausdorff dimension of the intersection of random translates of two Bedford-McMullen carpets and obtain a formula that extends the Kenyon-Peres formula from the self-similar to the self-affine setting.

Significance. If the central claims hold, the work supplies a dimension formula for random intersections in the self-affine setting and formulates a new problem on random matrix products. The relativised variational principle itself is a technical contribution that may find further use in dimension theory. The manuscript ships a clean statement of the dimension formula with no free parameters introduced by the authors.

major comments (2)
  1. [Application to Bedford-McMullen carpets] Application section (implicit in the abstract and the final paragraphs): the dimension formula for the random intersections is obtained by invoking Yin's theorem on the fibers of the factor maps associated with the Bedford-McMullen carpet projections. The text does not contain an explicit verification that these particular factor maps satisfy all hypotheses of Yin's result (continuity of the factor map, required properties of the entropy definitions, or the measure class). Because this equivalence is the direct link between the variational principle and the dimension formula, the omission is load-bearing for the main application.
  2. [Relativised variational principle] The relativised variational principle is stated for Feng-Huang weighted entropy, yet the manuscript gives no indication of the precise hypotheses under which the principle holds (e.g., compactness assumptions on the spaces or continuity requirements on the factor map). If these hypotheses are stricter than those needed for the carpet systems, the subsequent appeal to Yin's theorem cannot be made without additional argument.
minor comments (1)
  1. The abstract refers to 'a recent theorem of Yin' without a precise citation; the reference list should make the exact statement of Yin's result immediately locatable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the relativised variational principle and its application. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [Application to Bedford-McMullen carpets] Application section (implicit in the abstract and the final paragraphs): the dimension formula for the random intersections is obtained by invoking Yin's theorem on the fibers of the factor maps associated with the Bedford-McMullen carpet projections. The text does not contain an explicit verification that these particular factor maps satisfy all hypotheses of Yin's result (continuity of the factor map, required properties of the entropy definitions, or the measure class). Because this equivalence is the direct link between the variational principle and the dimension formula, the omission is load-bearing for the main application.

    Authors: We agree that an explicit verification strengthens the link to the main application. The factor maps are the canonical continuous projections from the symbolic coding space (a compact metric space) onto the Bedford-McMullen carpet; the dynamics are continuous, and the measures lie in the class required by Yin's theorem as set up in our preliminaries. We will add a dedicated paragraph in the application section (Section 4) that checks each hypothesis of Yin's result for these maps. revision: yes

  2. Referee: [Relativised variational principle] The relativised variational principle is stated for Feng-Huang weighted entropy, yet the manuscript gives no indication of the precise hypotheses under which the principle holds (e.g., compactness assumptions on the spaces or continuity requirements on the factor map). If these hypotheses are stricter than those needed for the carpet systems, the subsequent appeal to Yin's theorem cannot be made without additional argument.

    Authors: The relativised variational principle is established for continuous maps on compact metric spaces with a continuous factor map; these are the standing assumptions stated in the introduction and Section 2. The Bedford-McMullen systems satisfy them exactly. We will insert an explicit statement of these hypotheses at the beginning of the section containing the variational principle to remove any ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation combines new proof with external theorems

full rationale

The paper proves a relativised variational principle for Feng-Huang weighted entropy under factor maps as a new result. It then invokes an external recent theorem of Yin (distinct authors) to obtain a.e. equivalence on fibers, and applies this to extend the external Kenyon-Peres formula for the Hausdorff dimension of random intersections of Bedford-McMullen carpets. No quoted step reduces by construction to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central dimension formula depends on independently established external results rather than internal circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; insufficient detail available to populate the ledger.

pith-pipeline@v0.9.1-grok · 5615 in / 1274 out tokens · 31627 ms · 2026-06-27T23:23:40.805342+00:00 · methodology

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