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Multifractal aspects of $α$-expansions

Jörg Neunhäuserer

Abstract

In \cite{[NE]} we introduce $α$-expansions a real numbers in $(0,1]$, given by \[ \sum_{i=1}^{\infty}(α-1)^{i-1}α^{-(d_{1}+\dots+d_{i})}\] with $α>1$ and $d_{i}\in\mathbb{N}$ and discuss ergodic theoretical and dimension theoretical aspects of this expansions. In this sequel we study mutifractal aspects of this expansions.

Multifractal aspects of $α$-expansions

Abstract

In \cite{[NE]} we introduce -expansions a real numbers in , given by with and and discuss ergodic theoretical and dimension theoretical aspects of this expansions. In this sequel we study mutifractal aspects of this expansions.

Paper Structure

This paper contains 5 sections, 11 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Proof of theorem 2.1
  4. Proof of theorem 2.2
  5. Proof of theorem 2.3

Key Result

Theorem 2.1

We have

Figures (2)

  • Figure 1: The Khintchine-spectrum $\kappa$ for $\alpha\in\{3/2,2,3\}$
  • Figure 2: The dimension-spectrum $d(\mu)$ for $\alpha\in\{3/2,2,3\}$

Theorems & Definitions (11)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Theorem 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Corollary 4.1
  • Theorem 4.1
  • Proposition 5.1
  • ...and 1 more