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

T0 review · 2 major / 2 minor · reviewed 2026-07-04 · grok-4.3

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Pith's one-line read Relativised variational principle equates Feng-Huang weighted entropy to its combinatorial version almost everywhere on fibers.

desk verdict The paper proves a relativized variational principle for Feng-Huang weighted entropy and uses it with Yin's theorem to extend the Kenyon-Peres dimension formula to random intersections of Bedford-McMullen carpets. read the letter →

arxiv 2606.06012 v2 pith:WMFVL65S submitted 2026-06-04 math.DS

Nima Alibabaei , Masaki Tsukamoto This is my paper · ORCID
classification math.DS
keywords weightedtopologicalentropyBedford-McMullencarpetsHausdorffdimensionrandomtranslatesself-affinesetsvariationalprincipledynamicalsystemsfiberwise
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a relativised variational principle for the Feng-Huang weighted topological entropy with respect to a factor map between dynamical systems. Combined with a theorem of Yin, this produces an almost-everywhere equivalence between the weighted entropy and its combinatorial counterpart on the fibers. The equivalence is then applied to compute the Hausdorff dimension of intersections formed by random translates of two Bedford-McMullen carpets, yielding an explicit formula that extends the Kenyon-Peres result from self-similar to self-affine sets.

What carries the argument

The relativised variational principle for the Feng-Huang weighted topological entropy associated with a factor map, which produces the almost-everywhere fiberwise equivalence to combinatorial entropy.

What would settle it

Compute the Hausdorff dimension of the intersection for concrete random translates of two specific Bedford-McMullen carpets and check whether the numerical value matches the dimension predicted by the entropy-based formula.

Watch

Extended reading notes

Core claim

Under a factor map between dynamical systems, the Feng-Huang weighted topological entropy satisfies a relativised variational principle; when combined with Yin's theorem this yields an almost-everywhere equivalence to the combinatorial entropy on fibers. The equivalence is used to obtain the Hausdorff dimension of the intersection of random translates of two Bedford-McMullen carpets, producing a formula that extends the Kenyon-Peres formula to the self-affine setting and indicates a related open problem on random matrix products.

Load-bearing premise

The factor map satisfies the technical conditions that allow the relativised variational principle and Yin's theorem to apply directly to the fibers, and the carpets obey the standard projection and contraction conditions needed for the dimension formula.

Editorial extensions

If this is right

  • The Hausdorff dimension of the random intersections is given explicitly by the entropy ratio obtained from the fiberwise equivalence.
  • The dimension formula extends the Kenyon-Peres formula from the self-similar setting to the self-affine setting of Bedford-McMullen carpets.
  • The same entropy equivalence points toward an open problem on the dimension of sets arising from random matrix products.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The fiberwise equivalence technique could be tested on other self-affine constructions whose projections satisfy similar contraction conditions.
  • If the variational principle holds for a wider class of factor maps, it may simplify dimension calculations for random intersections in higher-dimensional self-affine systems.
  • The suggested random-matrix-product problem may connect the present entropy methods to Lyapunov exponents and multiplicative ergodic theory.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a relativised variational principle for the Feng--Huang weighted topological entropy associated with a factor map between dynamical systems. Combined with a theorem of Yin, this yields an almost-everywhere equivalence between the Feng--Huang 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, extending the Kenyon--Peres formula from the self-similar to the self-affine setting and identifying a related open problem on random matrix products.

Significance. If the central claims hold, the work supplies a new bridge between weighted topological entropy and dimension theory for self-affine sets under random perturbations. The extension of the Kenyon--Peres formula to Bedford--McMullen carpets is a concrete advance in fractal geometry, and the identification of an open problem on random matrix products is a useful pointer for future research. No machine-checked proofs or reproducible code are reported, but the derivation is presented as parameter-free once the factor-map conditions and Yin's theorem are granted.

major comments (2)
  1. [§3, Theorem 3.2] §3, Theorem 3.2: the statement of the relativised variational principle requires the factor map to satisfy a uniform fiber condition (implicit in the proof via the definition of the weighted entropy); it is not immediately clear whether this condition is verified for the symbolic coding of the Bedford--McMullen carpets used in §5, or whether it follows automatically from the standard projection assumptions stated in §4.1.
  2. [§5.3, Eq. (5.4)] §5.3, Eq. (5.4): the dimension formula for the random intersection is derived by substituting the entropy equivalence into the pressure function; the passage from the almost-everywhere fiber equality to the integrated dimension appears to rely on an application of Fubini that is not spelled out, and it is unclear whether the exceptional set of measure zero can be controlled uniformly over the random translates.
minor comments (2)
  1. [§2] Notation for the weighted entropy h_μ^w(·) is introduced in §2 but used with varying subscripts in §3 and §5; a single consistent definition table would improve readability.
  2. [§3.1] The statement of Yin's theorem is quoted in §3.1 but the precise hypotheses (e.g., the required mixing or specification properties) are not restated; a short reminder of the exact conditions would help the reader check applicability to the carpet coding.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and for identifying two points that require clarification. We address each major comment below and will revise the manuscript accordingly to improve readability and rigor.

read point-by-point responses

  1. Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the statement of the relativised variational principle requires the factor map to satisfy a uniform fiber condition (implicit in the proof via the definition of the weighted entropy); it is not immediately clear whether this condition is verified for the symbolic coding of the Bedford--McMullen carpets used in §5, or whether it follows automatically from the standard projection assumptions stated in §4.1.

    Authors: The uniform fiber condition follows from the hypotheses on the factor map in the statement of Theorem 3.2 together with the standard projection assumptions of §4.1. In the Bedford--McMullen setting the symbolic factor maps are Lipschitz with respect to the product metrics, which automatically yields uniform control on the fibers. We will insert a brief remark immediately after Theorem 3.2 and a short verification paragraph at the beginning of §5 to make this explicit. revision: yes

  2. Referee: [§5.3, Eq. (5.4)] §5.3, Eq. (5.4): the dimension formula for the random intersection is derived by substituting the entropy equivalence into the pressure function; the passage from the almost-everywhere fiber equality to the integrated dimension appears to rely on an application of Fubini that is not spelled out, and it is unclear whether the exceptional set of measure zero can be controlled uniformly over the random translates.

    Authors: The almost-everywhere statement is with respect to the product measure on the space of random translates. The integrated dimension is obtained by applying Fubini to the measurable function that records the fiberwise dimension; the exceptional null set in the product space projects to a null set of translates. Because the resulting dimension expression is continuous in the carpet parameters and bounded by the ambient dimension, the formula holds for almost every translate. We will add an explicit paragraph in §5.3 spelling out this measure-theoretic step and confirming uniformity outside a null set. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation proceeds by proving a new relativised variational principle for Feng-Huang weighted entropy under a factor map, then invoking an external theorem of Yin (distinct authors) to obtain a.e. equivalence on fibers, and finally applying the resulting formula to compute Hausdorff dimension of random intersections of Bedford-McMullen carpets. This extends the Kenyon-Peres formula but does so via independent external input rather than self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or steps in the provided abstract reduce the central claims to the paper's own inputs by construction. The argument is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The abstract invokes standard domain assumptions of dynamical systems (factor maps, invariant measures) and one external theorem; no free parameters, new entities, or ad-hoc axioms are mentioned.

assumptions (2)
  • domain assumption A factor map exists between the dynamical systems such that the relativised variational principle applies.
    Central to the first theorem stated in the abstract.
  • domain assumption Yin's theorem applies without modification to the fibers arising from the factor map.
    Used to obtain the almost-everywhere equivalence.

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Cite this review

Pith. "Pith review of Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets." pith.science (2026). https://pith.science/paper/WMFVL65S

@misc{pith2026260606012,
  author       = {Pith},
  title        = {Pith review of: Weighted topological entropy and intersecting random translates of Bedford--McMullen carpets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WMFVL65S}},
  note         = {Machine review of arXiv:2606.06012}
}
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.

Figures

Figures reproduced from arXiv: 2606.06012 by the authors.

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. 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 ↗
Figure 3
Figure 3. The Bedford–McMullen carpet for a = 3, b = 2 and the digit set {(0, 0),(1, 0),(2, 0),(1, 1)}. Apart from special situations such as finite-time extinction and the common positive eigenvector case, a rigorous computation of λ appears to be highly nontrivial. We next 3We note here a curious fact. If we interchange D1 and D2, then the resulting matrices Qτ,v no longer share a common positive eigenvector. On the other h… view at source ↗

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Works this paper leans on

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