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Hausdorff dimension of the limit sets of Tree Iterated Function Systems

Hiromichi Ono

Abstract

We investigate Tree Iterated Function Systems (TIFSs), which we introduce in this paper. TIFSs are the generalizations of Iterated Function Systems in which we take the maps independently at each step and each block. In this paper, we give the definition of TIFSs and the limit sets of them. We show a formula for the Hausdorff dimension of the limit sets of TIFSs, which is a generalization of Bowen's formula. Moreover, we give an example which emphasizes the difference between TIFSs and non-autonomous IFSs.

Hausdorff dimension of the limit sets of Tree Iterated Function Systems

Abstract

We investigate Tree Iterated Function Systems (TIFSs), which we introduce in this paper. TIFSs are the generalizations of Iterated Function Systems in which we take the maps independently at each step and each block. In this paper, we give the definition of TIFSs and the limit sets of them. We show a formula for the Hausdorff dimension of the limit sets of TIFSs, which is a generalization of Bowen's formula. Moreover, we give an example which emphasizes the difference between TIFSs and non-autonomous IFSs.
Paper Structure (5 sections, 17 theorems, 102 equations)

This paper contains 5 sections, 17 theorems, 102 equations.

Table of Contents

  1. Introduction and the main results
  2. Tree Iterated Function Systems
  3. Constructing a probability measure
  4. Proof of theorem \ref{['Thm:Main']}
  5. Example

Key Result

Theorem 1.1

Let $\Psi=\{\phi_\tau\}_{\tau\in T\setminus\{\emptyset\}}$ be a conformal TIFS and $J$ be the limit set. Let $t\ge0$. Then, we have the following. Especially, if the condition Eq:main_cond1 in (ii) is satisfied, then $\dim_H(J)=\beta^*(\Psi)$.

Theorems & Definitions (40)

  • Theorem 1.1: Theorem \ref{['Thm:Main']}
  • Theorem 1.2: see Theorems \ref{['Eg:Dim']}, \ref{['Eg:Zn']}
  • Definition 2.1
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 30 more