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New results on embeddings of self-similar sets via renormalization

Amir Algom, Michael Hochman, Meng Wu

Abstract

For self-similar sets $X,Y\subseteq \mathbb{R}$, we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of $X,Y$ have algebraic contraction ratios, and also for arbitrary $Y$ when the maps defining $X$ have algebraic-contraction ratios and there are sufficiently many of them relative to the number of maps defining $Y$. We also show that it holds for a.e. choice of the corresponding contraction ratios, and obtain bounds on the packing dimension of the exceptional parameters. Key ingredients in our argument include a new renormalization procedure in the space of embeddings, and Orponen's projection Theorem for Assouad dimension (2021).

New results on embeddings of self-similar sets via renormalization

Abstract

For self-similar sets , we obtain new results towards the affine embeddings conjecture of Feng-Huang-Rao (2014), and the equivalent weak intersections conjecture. We show that the conjecture holds when the defining maps of have algebraic contraction ratios, and also for arbitrary when the maps defining have algebraic-contraction ratios and there are sufficiently many of them relative to the number of maps defining . We also show that it holds for a.e. choice of the corresponding contraction ratios, and obtain bounds on the packing dimension of the exceptional parameters. Key ingredients in our argument include a new renormalization procedure in the space of embeddings, and Orponen's projection Theorem for Assouad dimension (2021).

Paper Structure

This paper contains 21 sections, 29 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Self-similar sets, and simplifying assumptions
  4. Assouad dimension
  5. Magnification and CP-distributions
  6. Proof of Theorems \ref{['thm:main-half']}
  7. Renormalization on the space $\mathcal{E}$
  8. The space of affine maps
  9. Dyadic cells in $\mathcal{A}$
  10. The renormalization maps $M_{D,\mathop{\mathrm{\textbf{i}}}\nolimits}$
  11. Computation of the renormalization map
  12. Limiting behavior of renormalization
  13. Proof of Theorem \ref{['thm:main']}
  14. Amplifying Assouad dimension to Hausdorff dimension
  15. Proof of Theorem \ref{['thm:main']}
  16. ...and 6 more sections

Key Result

Theorem 1.2

Conjecture conj:main holds when all the scaling constants $\alpha_{i},\beta_{j}$ are algebraic.

Theorems & Definitions (34)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • ...and 24 more