Critical exponent of a word

2020 Mathematics Subject Classification: Primary: 68R15 [MSN][ZBL]

For a finite or infinite word over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3.

If $w$ is an infinite word over the finite alphabet $A$ and $x$ is a finite word over $A$, then $x$ is said to occur in $w$ with exponent $\alpha$, for positive real $\alpha$, if there is a factor $y$ of $w$ with $y = x^a x_p$ where $x_p$ is a prefix of $x$, $a$ is the integer part of $\alpha$, and the length $|y| \ge \alpha |x|$|: we say that $y$ is an $\alpha$-power. The word $w$ is $\alpha$-power-free if it contains no factors which are $\alpha$-powers.

The critical exponent for $w$ is the supremum of the $\alpha$ for which $w$ has $\alpha$-powers, or equivalently the infimum of the $\alpha$ for which $w$ is $\alpha$-power-free.

Examples

• The critical exponent of the Fibonacci word is $\frac{5+\sqrt5}{2} = 3.618\ldots$.
• The critical exponent of the Thue–Morse sequence is 2. The word contains arbitrarily long squares, but in any factor $xxb$ the letter $b$ is not a prefix of $x$.

Properties

• The critical exponent can take any real value greater than 1.
• The critical exponent of a morphic word over a finite alphabet is an algebraic number of degree at most the size of the alphabet.

Repetition threshold

The repetition threshold of an alphabet $A$ of $n$ letters is the minimum critical exponent of infinite words over $A$: clearly this value $RT(n)$ depends only on $n$. For $n=2$, any binary word of length four has a factor of exponent 2, and since the critical exponent of the Thue–Morse sequence is 2, the repetition threshold for binary alphabets is $RT(2) = 2$. It is known that $RT(3) = 7/4$, $RT(4) = 7/5$ and that for $n \ge 5$ we have $RT(n) \ge 1 + 1/(n-1)$. It is conjectured that the latter is the true value, and this has been established for $5 \le n \le 14$ and for $n \ge 33$.

References

• Allouche, Jean-Paul; Shallit, Jeffrey; "Automatic Sequences: Theory, Applications, Generalizations", Cambridge University Press (2003) ISBN 978-0-521-82332-6 Zbl 1086.11015
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; "Combinatorics on words. Christoffel words and repetitions in words", CRM Monograph Series 27 American Mathematical Society (2009) ISBN 978-0-8218-4480-9 Zbl 1161.68043
• Krieger, Dalia; On critical exponents in fixed points of non-erasing morphisms pp. 280–291 in Oscar H. Ibarra, Zhe Dang (edd.) "Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006", Lecture Notes in Computer Science 4036 , Springer (2006) ISBN 3-540-35428-X Zbl 1227.68074
• Krieger, D.; Shallit, J.; Every real number greater than one is a critical exponent, Theor. Comput. Sci., 381 (2007), pp. 177–182 DOI 10.1016/j.tcs.2007.04.037
• Lothaire, M.; "Algebraic combinatorics on words", Encyclopedia of Mathematics and Its Applications 90 Cambridge University Press, (2011) ISBN 978-0-521-18071-9 Zbl 1221.68183
• Pytheas Fogg, N.; "Substitutions in dynamics, arithmetics and combinatorics", Lecture Notes in Mathematics 1794 Springer (2002) ISBN 3-540-44141-7 Zbl 1014.11015