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The orbit of a $β$-transformation cannot lie in a small interval

DoYong Kwon

TL;DR

This paper effectively determines the maximal domain of , and explicitly specifies all possible minimal intervals containing -orbits.

Abstract

For $β>1$, let $T_β:[0,1]\rightarrow [0,1)$ be the $β$-transformation. We consider an invariant $T_β$-orbit closure contained in a closed interval with diameter $1/β$, then define a function $Ξ(α,β)$ by the supremum of such $T_β$-orbit with frequency $α$ in base $β$, i.e., the maximum value in the $T_β$-orbit closure. This paper effectively determines the maximal domain of $Ξ$, and explicitly specifies all possible minimal intervals containing $T_β$-orbits. For Addendum: The paper mentioned in the title is completed by this Addendum.

The orbit of a $β$-transformation cannot lie in a small interval

TL;DR

This paper effectively determines the maximal domain of , and explicitly specifies all possible minimal intervals containing -orbits.

Abstract

For , let be the -transformation. We consider an invariant -orbit closure contained in a closed interval with diameter , then define a function by the supremum of such -orbit with frequency in base , i.e., the maximum value in the -orbit closure. This paper effectively determines the maximal domain of , and explicitly specifies all possible minimal intervals containing -orbits. For Addendum: The paper mentioned in the title is completed by this Addendum.

Paper Structure

This paper contains 10 sections, 16 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Definition of $\Xi$.
  3. Sturmian and Christoffel words.
  4. Domain of $\Xi$.
  5. (\ref{['GenericCase1-b']}) $\mathbf{a^{k+1}<v<ba^k}$.
  6. (\ref{['GenericCase1-c']}) $\mathbf{v>ba^k}$.
  7. (\ref{['GenericCase3-a']}) $\mathbf{xy=ab, v>uab}$, or $\mathbf{xy=ba, v<uba}$.
  8. (\ref{['GenericCase3-b']}) $\mathbf{xy=ab, v<uab}$, or $\mathbf{xy\leq aa}$.
  9. (\ref{['GenericCase3-c']})
  10. Diameters of $\overline{T}_\beta$-orbits.

Key Result

Theorem 1.1

Let $\beta\geq2$ be an integer and $\xi$ be an irrational number. Suppose that $s\leq\{\xi \beta^n\}\leq t$ for every integer $n\geq0$. Then $t-s$ cannot be smaller than $1/\beta$. Furthermore, $s\leq\{\xi \beta^n\}\leq s+1/\beta$ for every integer $n\geq0$ if and only if $\xi=\lfloor\xi\rfloor+ (w)

Theorems & Definitions (30)

  • Theorem 1.1: BD
  • Proposition 2.1
  • proof
  • Example 1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1: Pa
  • ...and 20 more