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Assouad type dimensions of infinitely generated self-conformal sets

Amlan Banaji, Jonathan M. Fraser

Abstract

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.

Assouad type dimensions of infinitely generated self-conformal sets

Abstract

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of conformal contractions. Our focus is on the Assouad type dimensions, which give information about the local structure of sets. Under natural separation conditions, we prove a formula for the Assouad dimension and prove sharp bounds for the Assouad spectrum in terms of the Hausdorff dimension of the limit set and dimensions of the set of fixed points of the contractions. The Assouad spectra of the family of examples which we use to show that the bounds are sharp display interesting behaviour, such as having two phase transitions. Our results apply in particular to sets of real or complex numbers which have continued fraction expansions with restricted entries, and to certain parabolic attractors.
Paper Structure (13 sections, 28 theorems, 108 equations, 1 figure)

This paper contains 13 sections, 28 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Structure of paper and summary of results
  4. Setting and preliminaries
  5. Notation and notions of dimension
  6. Conformal iterated function systems
  7. Assouad dimension of the limit set
  8. Bounds for the Assouad spectrum
  9. The bounds
  10. Consequences of Theorem \ref{['t:mainasp']}
  11. Sharpness of the bounds and attainable forms of Assouad spectra
  12. Applications: continued fractions and parabolic IFSs
  13. Further work

Key Result

Lemma 2.3

For any CIFS and any $\lambda > 0$ there are only finitely many $w \in I^*$ with $||S_w'|| \geqslant \lambda$.

Figures (1)

  • Figure 1: Assouad spectra of the sets in Theorem \ref{['t:sharp']} for $p=1.8$ and $h \approx 0.5$. In black: the upper bound (attained when $t=p+1$). In brown: the lower bound (attained when $t\geqslant p+h^{-1}$). In dashed blue: some more choices of $t \in (p+1,p+h^{-1})$. In dotted green: the case $t=2p$, which is also the dimension of the continued fraction set from Proposition \ref{['p:ctdspaced']}.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • proof
  • ...and 51 more