Tangents and slices of self-affine carpets

Abstract

We study the fine scaling properties of planar self-affine carpets. For Gatzouras--Lalley carpets, we give a precise formula for maximal Hausdorff dimension of a tangent in terms of the Hausdorff dimension of the projection and the Assouad dimension of the corresponding vertical slice. Using regularity properties for the Assouad dimension of non-autonomous self-similar sets, this implies that the set of points with tangents that are as large as possible has full Hausdorff measure, at the critical exponent. On the other hand, we give an explicit example of a Bara\'nski carpet for which the Hausdorff dimension of the set of points for which there exists a maximal tangent has Hausdorff dimension strictly less than the Hausdorff dimension of the original carpet.