A note on symmetries of rich sequences with minimum critical exponent

TL;DR

This paper explores the high symmetry properties of sequences with minimal critical exponent, showing they are G-rich with respect to groups generated by multiple antimorphisms, extending the concept of palindrome richness.

Contribution

It introduces the notion of G-richness for sequences with minimal critical exponent, generalizing palindrome richness to groups generated by multiple antimorphisms.

Findings

Thue-Morse sequence has minimum critical exponent among binary sequences

Binary rich sequences with minimum critical exponent are G-rich

Ternary rich sequences with minimum critical exponent are G-rich

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.