Slicing theorems and rigidity phenomena for self‐affine carpets

Abstract

Let F be a Bedford–McMullen carpet defined by independent exponents. We prove that dim¯Bfalse(ℓ∩Ffalse)⩽maxfalse{dim∗F−1,0false} for all lines ℓ not parallel to the principal axes, where dim∗ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford–McMullen carpets, that is, carpets F and E 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. (2) 189 (2019) 319–391] and Wu [Ann. of Math. (2) 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].

Improved versions of some Furstenberg type slicing Theorems for self-affine carpets

“…and that these inequalities are strict unless F is Ahlfors regular. So, for Ahlfors regular carpets the results of [1] are optimal for all notions of dimension previously discussed. However, in some sense "most" Bedford-McMullen carpets are not Ahlfors regular [6,Chapter 4].…”

Assouad dimension of planar self-affine sets

“…The classical Marstrand's projection theorem [49] for Hausdorff dimension states that, given a Borel set X ⊂ R 2 , we have dim H (proj V ⊥ (X)) = min{1, dim H (X)} for Lebesgue almost all V ∈ RP 1 . For a general class of self-affine sets, Bárány, Hochman, and Rapaport [7] have proved the above result for all V ∈ RP 1 .…”

“…It is therefore interesting to ask whether the slices are small also on other self-affine sets. For a Bedford-McMullen carpet X having logarithmically incommensurable contraction ratios, Algom [1] proved that the Hausdorff dimension of any slice not parallel to the principal axes is bounded above by max{0, dim A (X) − 1}. Besides the following theorem, we are not aware of any results of this type for general classes of non-carpet self-affine sets.…”

“…We denote the Hausdorff dimension by dim H and the collection of all lines through the origin by RP 1 . The classical Marstrand's projection theorem [49] for Hausdorff dimension states that, given a Borel set X ⊂ R 2 , we have dim H (proj V ⊥ (X)) = min{1, dim H (X)} for Lebesgue almost all V ∈ RP 1 .…”

“…The following theorem can therefore be considered as a manifestation of the rigid structure of self-affine sets. The set X F ⊂ RP 1 is the collection of all Furstenberg directions, i.e. the limits of all directions of stronger contraction, and under the assumptions of the theorem, it satisfies dim H (X F ) > 0.…”