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Tangents and slices of self-affine carpets

Antti Käenmäki, Alex Rutar

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 Barański 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.

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 Barański 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.

Paper Structure

This paper contains 21 sections, 25 theorems, 150 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Assouad dimensions
  3. Main results and outline of paper
  4. Notation
  5. Tangents and pointwise Assouad dimension
  6. Tangents and weak tangents
  7. Metric trees
  8. Regularity of non-autonomous self-similar sets
  9. Tangent structure and dimension of Gatzouras--Lalley carpets
  10. Gatzouras--Lalley and Barański carpets
  11. Defining the maps
  12. Dimensions of Gatzouras--Lalley carpets
  13. Regular points and interior words
  14. Approximate squares and symbolic slices
  15. Tangents of Gatzouras--Lalley carpets
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $K$ be a Gatzouras--Lalley carpet. Then On the other hand, for any $\mathop{\mathrm{dim_B}}\nolimits K\leq\alpha\leq\mathop{\mathrm{dim_A}}\nolimits K$, Moreover, if $\eta(K)$ satisfies the SSC, then for any $x\in K$,

Figures (5)

  • Figure 1: Gatzouras--Lalley
  • Figure 2: Barański
  • Figure 4: Gatzouras--Lalley
  • Figure 5: Barański
  • Figure 7: Two iterations of a Gatzouras--Lalley IFS within a cylinder, with a wide pseudo-cylinder in highlighted in blue and a tall pseudo-cylinder in red.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 3.1
  • Proposition 3.2: lg1992mac2011
  • ...and 33 more