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Hausdorff dimension of the parameters for $(α,β)$-transformations with the specification property

Mai Oguchi, Mao Shinoda

Abstract

In this paper we consider the specification property for $(α,β)$-shifts. When $α=0$, Schmeling shows that the set of $β>1$ for which the $β$-shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension\cite{Schmeling}. So it is natural to ask what happens when $α>0$. Buzzi shows that for fixed $α$ the set of $β>1$ for which the $(α,β)$-shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of $(α,β)$-shifts.

Hausdorff dimension of the parameters for $(α,β)$-transformations with the specification property

Abstract

In this paper we consider the specification property for -shifts. When , Schmeling shows that the set of for which the -shift has the specification property has the Lebesgue measure zero but has the full Hausdorff dimension\cite{Schmeling}. So it is natural to ask what happens when . Buzzi shows that for fixed the set of for which the -shift has the specification property has Lebesgue measure zero. Hence we consider the Hausdorff dimension of the parameter space of -shifts.
Paper Structure (5 sections, 4 theorems, 33 equations)

This paper contains 5 sections, 4 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symbolic dynamics
  4. $(\alpha,\beta)$-shifts
  5. Proof of the main theorem

Key Result

Theorem 1

Let $\ul{u}\in \mathbb{N}_0^{\mathbb{N}_0}$ satisfy $u_0=0$, $\ul{u}\preceq \sigma^n \ul{u}$ for all $n\geq 0$ and $\max_{i\in \mathbb{N}_0}u_i=:K<\infty$. Then we have where $\alpha(\beta)=(\beta-1)\sum_{j=0}^\infty \frac{u_j}{\beta^{j+1 }}$ and $\dim_H E$ is the Hausdorff dimension of $E$.

Theorems & Definitions (6)

  • Theorem 1
  • Definition 2.1: Specification
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2