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2-balanced sequences coding rectangle exchange transformation

Lubomíra Dvořáková, Zuzana Masáková, Edita Pelantová

TL;DR

An upper bound on factor and abelian complexity of a new class of ternary sequences that are 2-balanced is provided, which shows that the class contains sequences of any given letter frequencies.

Abstract

We define a new class of ternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We show that the class contains sequences of any given letter frequencies. We provide an upper bound on factor and abelian complexity of these sequences. Using the interpretation by rectangle exchange transformation, we prove that for almost all triples of letter frequencies, the upper bound on factor and abelian complexity is reached. The bound on factor complexity is given using a number-theoretical function which we compute explicitly for a class of parameters.

2-balanced sequences coding rectangle exchange transformation

TL;DR

An upper bound on factor and abelian complexity of a new class of ternary sequences that are 2-balanced is provided, which shows that the class contains sequences of any given letter frequencies.

Abstract

We define a new class of ternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We show that the class contains sequences of any given letter frequencies. We provide an upper bound on factor and abelian complexity of these sequences. Using the interpretation by rectangle exchange transformation, we prove that for almost all triples of letter frequencies, the upper bound on factor and abelian complexity is reached. The bound on factor complexity is given using a number-theoretical function which we compute explicitly for a class of parameters.
Paper Structure (8 sections, 10 theorems, 44 equations, 1 figure, 6 tables)

This paper contains 8 sections, 10 theorems, 44 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Colouring of sequences and balancedness
  4. On factor and abelian complexity
  5. Ternary 2-balanced sequences
  6. Ternary 2-balanced sequences as coding of rectangle exchange transformation
  7. Symbolic computation of factor complexity
  8. Comments

Key Result

Lemma 4

Let $\mathbf{u}$ be a 1-balanced sequence over $\{ a,b\}$, and ${\bf a}=1^{\omega}$ and ${\bf b} = b_0b_1b_2\dots$ be a 1-balanced sequence over $\{2,3\}$. Then the ternary sequence ${\bf v} = \mathrm{colour}(\mathbf{u}, {\bf a},{\bf b})$ is $2$-balanced.

Figures (1)

  • Figure 1: Transformation S from \ref{['eq:RectTrans']}

Theorems & Definitions (30)

  • Remark 1
  • Definition 2
  • Example 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Definition 6
  • Lemma 7
  • proof
  • ...and 20 more