Slicing theorems and rigidity phenomena for self-affine carpets

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Let (Formula presented.) be a Bedford–McMullen carpet defined by independent exponents. We prove that (Formula presented.) for all lines (Formula presented.) not parallel to the principal axes, where (Formula presented.) is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford–McMullen carpets, that is, carpets (Formula presented.) and (Formula presented.) such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self-affine measures on them are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing theorem for products of certain Cantor sets. This theorem is a generalization of the results of Shmerkin [Ann. of Math. (Formula presented.) 189 (2019) 319–391] and Wu [Ann. of Math. (Formula presented.) 189 (2019) 707–751], which proved Furstenberg's slicing conjecture [Problems in analysis (ed. R. C. Gunning; Princeton University Press, Princeton, NJ, 1970) 41–59].

Original languageEnglish
Pages (from-to)312-353
Number of pages42
JournalProceedings of the London Mathematical Society
Issue number2
StatePublished - 1 Aug 2020
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.


  • 11K55
  • 28A50
  • 28A80 (primary)
  • 28D05
  • 37C45 (secondary)

ASJC Scopus subject areas

  • General Mathematics


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