Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on symmetries of rich sequences with minimum critical exponent

Lubomíra Dvořáková, Edita Pelantová

TL;DR

This work investigates symmetry properties of sequences with minimal repetition exponents within their class, showing that high symmetry manifests as $G$-richness with respect to groups generated by multiple antimorphisms. It proves that the ternary rich sequence $\\mathbf{z}$ with minimal critical exponent is $G$-rich for $G=\\{I,R,S,RS\\}$ via a morphic-image representation and analysis of palindromic complexity, and connects binary minimal-exponent examples to $G$-richness through complementary symmetric Rote sequences and $G=\\{I,R,E,ER\\}$. The paper provides explicit morphic constructions and a generalized richness criterion (via the graph of symmetries $\\Gamma_n$ and related equalities) to certify $G$-richness, and ends with an open problem about extending the phenomenon to larger alphabets. It also places the results in the context of known repetition thresholds, namely binary $2+\\frac{\\sqrt{2}}{2}$ and ternary $1+\\frac{1}{3-\\mu}$ with $\\mu$ the real root of $x^3-2x^2-1$.

Abstract

Using three examples of sequences over a finite alphabet, we want to draw attention to the fact that these sequences having the minimum critical exponent in a given class of sequences show a large degree of symmetry, i.e., they are G-rich with respect to a group G generated by more than one antimorphism. The notion of G-richness generalizes the notion of richness in palindromes which is based on one antimorphism, namely the reversal mapping. The three examples are: 1) the Thue-Morse sequence which has the minimum critical exponent among all binary sequences; 2) the sequence which has the minimum critical exponent among all binary rich sequences; 3) the sequence which has the minimum critical exponent among all ternary rich sequences.

A note on symmetries of rich sequences with minimum critical exponent

TL;DR

This work investigates symmetry properties of sequences with minimal repetition exponents within their class, showing that high symmetry manifests as -richness with respect to groups generated by multiple antimorphisms. It proves that the ternary rich sequence with minimal critical exponent is -rich for via a morphic-image representation and analysis of palindromic complexity, and connects binary minimal-exponent examples to -richness through complementary symmetric Rote sequences and . The paper provides explicit morphic constructions and a generalized richness criterion (via the graph of symmetries and related equalities) to certify -richness, and ends with an open problem about extending the phenomenon to larger alphabets. It also places the results in the context of known repetition thresholds, namely binary and ternary with the real root of .

Abstract

Using three examples of sequences over a finite alphabet, we want to draw attention to the fact that these sequences having the minimum critical exponent in a given class of sequences show a large degree of symmetry, i.e., they are G-rich with respect to a group G generated by more than one antimorphism. The notion of G-richness generalizes the notion of richness in palindromes which is based on one antimorphism, namely the reversal mapping. The three examples are: 1) the Thue-Morse sequence which has the minimum critical exponent among all binary sequences; 2) the sequence which has the minimum critical exponent among all binary rich sequences; 3) the sequence which has the minimum critical exponent among all ternary rich sequences.
Paper Structure (7 sections, 7 theorems, 16 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The rich ternary sequence $\mathbf{z}$ with the minimum critical exponent as a morphic image of a fixed point
  4. Definition of generalized richness of sequences
  5. G-richness of the rich ternary sequence $\mathbf{z}$
  6. G-richness of the rich binary sequences with the minimum critical exponent
  7. Open problem

Key Result

Lemma 1

$f^n\psi = \psi \varphi^n$ for every $n \in \mathbb N$.

Figures (2)

  • Figure 1: The graph of symmetries ${\Gamma}_2(\mathbf{u})$ for $\mathbf{u}=\varphi^{\omega}(0)$ and the group $G=\{I,R\}$.
  • Figure 2: The graphs of symmetries ${\Gamma}_1(\mathbf{z})$ (left) and ${\Gamma}_2(\mathbf{z})$ (right) for $\mathbf{z}$ defined in \ref{['eq:Z']} and the group $G=\{I,R, S, RS\}$.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • Example 4
  • Definition 5
  • Definition 6
  • Remark 7
  • Example 8
  • ...and 8 more