View source for Critical exponent of a word

{{TEX|done}}{{MSC|68R15}}

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 $RT(n) = 1 + 1/(n-1)$.


====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}}
* Currie, James; Rampersad, Narad; "A proof of Dejean's conjecture" ''Math. Comp.'' '''80''' (2011), 1063-1070. {{DOI|10.1090/S0025-5718-2010-02407-X}}. {{MR|2772111}}  {{ZBL|1215.68192}}
* 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}}
* Rao, Michaël; "Last cases of Dejean’s conjecture". ''Theor. Comput. Sci.'' '''412''' (2011) 3010-3018.  {{DOI|10.1016/j.tcs.2010.06.020}}. {{ZBL|1230.68163}}