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Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections

Meysam Nassiri, Mojtaba Zareh Bidaki

Abstract

It is well known that a pair of compact sets in $\mathbb{R}^d$ ($d \in \mathbb{N}$) can be separated by small deformations if the sum of their Hausdorff dimensions is less than $d$. In this paper, we demonstrate that this dimension constraint is optimal for regular Cantor sets. Specifically, for any prescribed Hausdorff dimensions whose sum is greater than $d$, we construct classes of pairs of regular Cantor sets that exhibit $C^{1+α}$-stable intersections. Our method is geometrically flexible, enabling the construction of examples with arbitrarily small thickness in both projectively hyperbolic and nearly conformal regimes. These results also extend to the complex setting for holomorphic Cantor sets in $\mathbb{C}^d$. The proof relies on the ``covering criterion" for stable intersection introduced in the first part of this series \cite{NZ1}, which generalizes the ``recurrent compact set criterion" of Moreira-Yoccoz to higher dimensions.

Cantor sets in higher dimensions II: Optimal dimension constraint for stable intersections

Abstract

It is well known that a pair of compact sets in () can be separated by small deformations if the sum of their Hausdorff dimensions is less than . In this paper, we demonstrate that this dimension constraint is optimal for regular Cantor sets. Specifically, for any prescribed Hausdorff dimensions whose sum is greater than , we construct classes of pairs of regular Cantor sets that exhibit -stable intersections. Our method is geometrically flexible, enabling the construction of examples with arbitrarily small thickness in both projectively hyperbolic and nearly conformal regimes. These results also extend to the complex setting for holomorphic Cantor sets in . The proof relies on the ``covering criterion" for stable intersection introduced in the first part of this series \cite{NZ1}, which generalizes the ``recurrent compact set criterion" of Moreira-Yoccoz to higher dimensions.
Paper Structure (8 sections, 7 theorems, 118 equations, 8 figures)

This paper contains 8 sections, 7 theorems, 118 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Covering conditions
  4. Covering in Linear Groups
  5. Proof of main theorems
  6. Proof of Theorem \ref{['thm: main1']}
  7. Proof of Theorem \ref{['thm: main2']}
  8. Homogeneous Cantor sets on the real line

Key Result

Theorem 1

Let $d \in \mathbb{N}$ and $p,q \in (0,d)$ with $p+q>d$. Then there exists a pair of regular Cantor sets $(K,K')$ in $\mathbb{R}^d$ with $\operatorname{dim_{H}}(K)= p$ and $\operatorname{dim_{H}}(K')=q$ such that the pair $(K,K')$ has $\mathcal{C}^{1+\alpha}$-stable intersection ($\alpha>0$).

Figures (8)

  • Figure 1: Dimension pairs $(\operatorname{dim_{H}}(K), \operatorname{dim_{H}}(K'))$. Shaded region: pairs previously achieved by blender-type Cantor sets. Light gray region: new dimension pairs accessed via Theorems \ref{['thm: main1']} and \ref{['thm: main2']}.
  • Figure 2: Cantor set $K_d$ in dimension $d=2$, first step approximation.
  • Figure 3: The first step approximation of Cantor set $K'$ on the left, and its zoomed building blocks on the right. The blue rectangles show $K'_d$ in dimension $d=2$.
  • Figure 4: The first step approximation of Cantor set $K'$ on the left, and its zoomed building blocks on the right. The blue rectangles show $K'_A$ in dimension $d=2$, and for some diagonal matrix $A$.
  • Figure 5: First step approximation of the Cantor set $K_1$.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Conjecture 1.1
  • Definition 2.1: $\mathcal{C}^{1+\alpha}$ neighborhood of a regular Cantor set
  • Definition 2.2: Stable intersection
  • Definition 2.3: Bunched Cantor sets
  • Definition 3.1: Covering conditions
  • Lemma 3.2: Quantitative covering near $\mathrm{Id}$
  • proof : Proof of Lemma \ref{['lem: generating a convering in SL with d^2 maps']}
  • Proposition 3.3
  • ...and 8 more