Self-similar sets and Lipschitz graphs

TL;DR

This work quantifies the distinction between rectifiable and purely unrectifiable 1-sets in the plane by linking self-similar attractors of iterated function systems to Lipschitz-graph parametrizations. It develops a constructive graph-construction framework and leverages projection theory, Favard length, and ergodic methods to show that, under open set conditions, large subsets of 1D IFS attractors lie on Lipschitz graphs with explicitly controlled Lipschitz constants; the results hold in rotation-free and general rotational settings, with concrete bounds illustrated on the 4-corner Cantor set. The main contributions include a Lipschitz-graph intersection theorem for 1D IFS attractors, explicit Lip-constants depending on dimension gaps, and novel, constructive approaches to identify good projection directions across scales. These findings enhance understanding of rectifiability and provide tools for parametrizing fractal 1-sets via Lipschitz graphs, with potential applications in geometric measure theory and projection problems.

Abstract

We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph.

Figures (8)

• Figure 1: The first two iterations in the definition of $\mathcal{C}_{6}$
• Figure 2: From left to right, the images of $E_1^1$, $E_1^2$, and $E_1^3$ are shown in black. The numbering of cubes is indicated for $E_1^1$ and $E_1^2$. In $E_1^1$, some of the corners are labelled. The cubes that are not selected for each $E_1^m$ are shown in gray.
• Figure 3: From left to right, the graphs $\Gamma^1_1$, $\Gamma^1_2$, and $\Gamma^1_3$ are shown in black over the sets $E^1_1$, $E^1_2$, and $E^1_3$, respectively, shown in gray. In each image, the vectors $\tau_1^1$ and $\tau^1_2$ are indicated with dashed lines and labelled.
• Figure 4: From left to right, the images of $\Gamma^2_1$ and $\Gamma^2_2$ (top row), then $\Gamma^3_1$ and $\Gamma^3_2$ (bottom row), the graphs of the functions that limit to $g^2$ (top) and $g^3$ (bottom) are shown in black. Under each $\Gamma^m_n$, the set $E^m_n$ is shown in gray. In each image, the vectors $\tau_1^1$ and $\tau^1_2$ are indicated with dashed lines and labelled.
• Figure 5: From left to right, the images of $E^1_1$, $E^2_1$, and $E^3_1$ are shown. The numbering of cubes is indicated in $E^1_1$ and $E^2_1$. The omitted cubes are in gray.

Theorems & Definitions (78)

• Definition 2.1: Lipschitz functions
• Definition 2.2: Hausdorff measure and dimension
• Definition 2.3: Ahlfors regularity
• Proposition 2.4: Simple Venetian blind construction
• proof
• Definition 2.5: Dyadic cubes
• Definition 2.6: $\beta$-numbers
• Theorem 2.7: Analyst's Traveling Salesperson Theorem, Theorem 1 in jones1990rectifiable
• Proposition 2.8: Lipschitz images result
• proof