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Equicontractive weak separation property on the line does not imply convex finite type condition

Kevin G. Hare

Abstract

Let $\{S_1, S_2, \dots, S_n\}$ be an iterated function system on $\mathbb{R}$ with attractor $K$. It is known that if the iterated function system satisfies the weak separation property and $K = [0,1]$ then the iterated function system also satisfies the convex finite type condition. We show that the condition $K = [0,1]$ is necessary. That is, we give two examples of iterated function systems on $\mathbb{R}$ satisfying weak separation condition, and $0< \dim_H(K) < 1$ such that the IFS does not satisfy the convex finite type condition.

Equicontractive weak separation property on the line does not imply convex finite type condition

Abstract

Let be an iterated function system on with attractor . It is known that if the iterated function system satisfies the weak separation property and then the iterated function system also satisfies the convex finite type condition. We show that the condition is necessary. That is, we give two examples of iterated function systems on satisfying weak separation condition, and such that the IFS does not satisfy the convex finite type condition.
Paper Structure (13 sections, 1 theorem, 10 equations, 2 figures)

This paper contains 13 sections, 1 theorem, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of first example
  3. Overview of IFS
  4. Constructing the IFS
  5. The IFS satisfies WSP
  6. The IFS satisfies OSC${}_{\mathrm{co}}$ but not OSC
  7. The IFS does not satisfy FTC${}_{\mathrm{co}}$
  8. Example
  9. Construction of second example
  10. Construction of IFS
  11. The IFS satisfies WSP and not FTC${}_{\mathrm{co}}$, OSC, nor OSC${}_{\mathrm{co}}$
  12. The IFS satisfies FTC
  13. Example

Key Result

Corollary 1.1

Suppose the IFS $\mathcal{S}$ has self-similar set $[0,1]$. Then then following are equivalent:

Figures (2)

  • Figure 2.1: Level $n$ cylinders for Example \ref{['ssec:OSC example']}
  • Figure 3.2: Level $n$ cylinders for Example \ref{['ssec:FTC example']}

Theorems & Definitions (1)

  • Corollary 1.1: Corollary 4.6 of HHR21