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Critical exponent of binary words with few distinct palindromes

L'ubomíra Dvořáková, Pascal Ochem, Daniela Opočenská

Abstract

We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with 18 and 20 palindromes happen to be morphic images of the fixed point of the morphism $\texttt{0}\mapsto\texttt{01}$, $\texttt{1}\mapsto\texttt{21}$, $\texttt{2}\mapsto\texttt{0}$.

Critical exponent of binary words with few distinct palindromes

Abstract

We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with 18 and 20 palindromes happen to be morphic images of the fixed point of the morphism , , .
Paper Structure (19 sections, 27 theorems, 37 equations, 1 table)

This paper contains 19 sections, 27 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fewest palindromes, least critical exponent, and factor complexity
  4. General result
  5. Exponential cases
  6. Polynomial cases
  7. The critical exponent of $\nu(\bf p)$ and $\mu(\bf p)$
  8. The infinite word $\bf p$
  9. Bispecial factors in $\bf p$
  10. The shortest return words to bispecial factors in $\bf p$
  11. The asymptotic critical exponent of $\bf p$
  12. The infinite word $\nu(\bf p)$
  13. Bispecial factors in $\nu(\bf p)$
  14. The shortest return words to bispecial factors in $\nu(\bf p)$
  15. The critical exponent of $\nu(\bf p)$
  16. ...and 4 more sections

Key Result

Theorem 1

There exists an infinite binary $\beta^+$-free word containing only $p$ palindromes for the following pairs $(p, \beta)$. Moreover, this list of pairs is optimal.

Theorems & Definitions (49)

  • Theorem 1
  • proof
  • Lemma 2
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 39 more