Critical exponent of a word

In mathematics and computer science, the critical exponent of a finite or infinite sequence of symbols 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 alphabet A and x is a finite word over A, then x is said to occur in w with exponent α, for positive real α, if there is a factor y of w with y = x^ax₀ where x₀ is a prefix of x, a is the integer part of α, and the length |y| = α |x|: we say that y is an α-power. The word w is α-power-free if it contains no factors which are β-powers for any β ≥ α.

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