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Hausdorff and box dimension bounds for subfractals induced by $S$-gap shifts

Elizabeth Sattler

Abstract

In this paper, we consider subsets of an attractor of an iterated function system in which each point is associated with an allowable word from an $S$-gap shift. The main result shows that bounds for the box dimension and Hausdorff dimension of a subfractal induced by an $S$-gap shift are given by the zeros of the upper and lower topological pressure functions.

Hausdorff and box dimension bounds for subfractals induced by $S$-gap shifts

Abstract

In this paper, we consider subsets of an attractor of an iterated function system in which each point is associated with an allowable word from an -gap shift. The main result shows that bounds for the box dimension and Hausdorff dimension of a subfractal induced by an -gap shift are given by the zeros of the upper and lower topological pressure functions.
Paper Structure (5 sections, 11 theorems, 39 equations)

This paper contains 5 sections, 11 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Topological pressure function for subshifts
  4. Main Theorem
  5. Acknowledgments

Key Result

Theorem 1

Let $h$, $H$ be the unique values such that $P(h) = 0 = \bar{P}(H)$. Then,

Theorems & Definitions (19)

  • Theorem
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • Corollary 3.5
  • Lemma 3.6
  • proof
  • ...and 9 more