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Cantor sets in higher dimension I: Criterion for stable intersections

Meysam Nassiri, Mojtaba Zareh Bidaki

TL;DR

The paper develops a higher dimensional extension of the Moreira–Yoccoz stable intersection theory for Cantor sets by introducing a $C^{1+\alpha}$ stable-intersection criterion under a mild bunching condition. It builds a renormalization framework and the bounded geometry/control of limit geometries to prove a finite-dimensional covering criterion that implies stability of intersections for a broad class of Cantor pairs in real and complex spaces. The authors also demonstrate explicit constructions of stably intersecting Cantor sets in any dimension, including cases with arbitrarily small Hausdorff dimension, and provide affine and holomorphic holonomy variants. This work advances the bifurcation and intersection theory of high‑dimensional Cantor sets and offers practical methods to produce explicit stably intersecting examples near conformal or affine limits.

Abstract

We study the geometry of dynamically defined Cantor sets in arbitrary dimensions, introducing a criterion for $\mathcal{C}^{1+α}$ stable intersections of such Cantor sets, under a mild bunching condition. This condition is naturally satisfied for perturbations of conformal Cantor sets and, in particular, always holds in dimension one. Our work extends the celebrated recurrent compact set criterion of Moreira and Yoccoz for stable intersection of Cantor sets in the real line to higher-dimensional spaces. Based on this criterion, we develop a method for constructing explicit examples of stably intersecting Cantor sets in any dimension. This construction operates in the most fragile and critical regimes, where the Hausdorff dimension of one of the Cantor sets is arbitrarily small and both Cantor sets are nearly homothetical. All results and examples are provided in both real and complex settings.

Cantor sets in higher dimension I: Criterion for stable intersections

TL;DR

The paper develops a higher dimensional extension of the Moreira–Yoccoz stable intersection theory for Cantor sets by introducing a stable-intersection criterion under a mild bunching condition. It builds a renormalization framework and the bounded geometry/control of limit geometries to prove a finite-dimensional covering criterion that implies stability of intersections for a broad class of Cantor pairs in real and complex spaces. The authors also demonstrate explicit constructions of stably intersecting Cantor sets in any dimension, including cases with arbitrarily small Hausdorff dimension, and provide affine and holomorphic holonomy variants. This work advances the bifurcation and intersection theory of high‑dimensional Cantor sets and offers practical methods to produce explicit stably intersecting examples near conformal or affine limits.

Abstract

We study the geometry of dynamically defined Cantor sets in arbitrary dimensions, introducing a criterion for stable intersections of such Cantor sets, under a mild bunching condition. This condition is naturally satisfied for perturbations of conformal Cantor sets and, in particular, always holds in dimension one. Our work extends the celebrated recurrent compact set criterion of Moreira and Yoccoz for stable intersection of Cantor sets in the real line to higher-dimensional spaces. Based on this criterion, we develop a method for constructing explicit examples of stably intersecting Cantor sets in any dimension. This construction operates in the most fragile and critical regimes, where the Hausdorff dimension of one of the Cantor sets is arbitrarily small and both Cantor sets are nearly homothetical. All results and examples are provided in both real and complex settings.

Paper Structure

This paper contains 30 sections, 35 theorems, 157 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The criterion
  3. About the proofs
  4. Preliminaries
  5. Basic notations
  6. Regular Cantor sets
  7. Bunched Cantor Sets
  8. The covering conditions
  9. Covering in linear groups
  10. Convergent geometries
  11. Existence and continuity of limits
  12. Strong bounded distortion
  13. Sequence of geometries
  14. Infinitesimal geometry of Cantor sets
  15. Configurations and renormalizations
  16. ...and 15 more sections

Key Result

Theorem 1

For every $d\in\mathbb{N}$, $\alpha\in(0,1)$ and $\epsilon>0$, there exists a pair of Cantor sets $(K, K')$ in $\mathbb{R}^d$ with $\mathcal{C}^{1+\alpha}$ stable intersection such that $\dim_{\rm{HD}}(K) <\epsilon$. Moreover, the Cantor sets $K,K'$ may be affine and arbitrarily close to the space o

Figures (8)

  • Figure 1: A Cantor set generated by four affine maps: initial approximations.
  • Figure 2: Sequence of maps satisfying (H0)-(H3) in Corollary \ref{['cor: control-of-shape']}, where the iterations of small balls remain almost ellipsoids.
  • Figure 3: $B_{\delta'}\left(\Phi_q(\mathcal{W})\right)$ satisfies strong covering condition with respect to the family $\mathcal{R}$. In this figure, ${\bf{u}}:= [\eta_{\hat{h}} \circ k^{\underline{\theta}}, \eta'_{\hat{h}'} \circ B_{\hat{h}, \hat{h}'}\circ k^{\underline{\theta}'}]$, ${\bf{a}}:=[k^{\underline{\theta}}, B_{\hat{h},\hat{h}'} \circ k^{\underline{\theta}'}]$ and ${\bf{a}'}: = [k^{\underline{\theta}(\hat{h}_1)}, B_{\hat{h}_1,\hat{h}'_1} \circ k^{\underline{\theta}'(\hat{h}'_1)}]$.
  • Figure 4: Cantor set $K_1'$, approximation in first and second steps.
  • Figure 5: Cantor set $K_1$, approximation in first step.
  • ...and 3 more figures

Theorems & Definitions (78)

  • Theorem 1
  • Remark 1.3
  • Theorem 2
  • Theorem 3: Covering criterion for stable intersection
  • Theorem 4
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 68 more