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Self-projective sets

Argyrios Christodoulou, Natalia Jurga

Abstract

Self-projective sets are natural fractal sets which describe the action of a semigroup of matrices on projective space. In recent years there has been growing interest in studying the dimension theory of self-projective sets, as well as progress in the understanding of closely related objects such as Furstenberg measures. The aim of this survey is twofold: first to motivate the study of these objects from several different perspectives and second to make the study of these objects more accessible for readers with expertise in iterated function systems.

Self-projective sets

Abstract

Self-projective sets are natural fractal sets which describe the action of a semigroup of matrices on projective space. In recent years there has been growing interest in studying the dimension theory of self-projective sets, as well as progress in the understanding of closely related objects such as Furstenberg measures. The aim of this survey is twofold: first to motivate the study of these objects from several different perspectives and second to make the study of these objects more accessible for readers with expertise in iterated function systems.
Paper Structure (17 sections, 11 theorems, 33 equations, 1 figure)

This paper contains 17 sections, 11 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Limit sets of Fuchsian semigroups
  3. Rauzy gasket
  4. Connection to the Furstenberg measure
  5. Self-projective sets for subsets of $\mathrm{SL}(2,\mathbb{R})$
  6. Conformality/ non-conformality.
  7. Contraction properties
  8. Algebraic properties
  9. Dimension of sets and measures
  10. Dimension of the Furstenberg measure
  11. Dimension of self-projective sets
  12. Examples
  13. Problems
  14. The higher dimensional setting
  15. Contraction properties
  16. ...and 2 more sections

Key Result

Theorem 1.2

If $G$ is an irreducible semigroup that contains a proximal matrix, then its limit set is the closure (in $\mathbb{R}\mathbb{P}^{d-1}$) of set of all attracting fixed points of proximal elements of $G$.

Figures (1)

  • Figure 1: The Rauzy gasket contained in the standard 2-simplex $\Delta$

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1: Uniform hyperbolicity
  • Definition 2.2: Ellipticity
  • Definition 2.3: Semidiscrete
  • Definition 2.5
  • Definition 2.6: Diophantine property
  • Theorem 2.9
  • Definition 2.10: Zeta function and critical exponent
  • Theorem 2.11
  • ...and 19 more