# Infinite words¶

Infinite words

Methods that are not in Sage (for now!)

AUTHORS:

• Sébastien Labbé, 2016

EXAMPLES:

sage: from slabbe.infinite_word import derived_sequence
sage: w = words.ThueMorseWord()
sage: derived_sequence(w, w[:1])
word: 0120210121020120210201210120210121020121...

slabbe.infinite_word.derived_sequence(self, u, coding=False)

Return the derived sequence of according to the return words to a factor of self.

INPUT:

• u – finite word, the length of the nonempty prefix
• coding – boolean (default: False), whether to include the return word coding dictionnary

EXAMPLES:

sage: from slabbe.infinite_word import derived_sequence
sage: w = words.ThueMorseWord()
sage: derived_sequence(w, w[:1])
word: 0120210121020120210201210120210121020121...
sage: derived_sequence(w, w[:2])
word: 0123013201232013012301320130123201230132...
sage: derived_sequence(w, w[:3])
word: 0123013201232013012301320130123201230132...


With the return word coding:

sage: w = words.ThueMorseWord()
sage: derived, D = derived_sequence(w, w[:1], True)
sage: derived
word: 0120210121020120210201210120210121020121...
sage: D
{word: 0: 2, word: 01: 1, word: 011: 0}


It gets into a cycle of length 1:

sage: words.ThueMorseWord()
word: 0110100110010110100101100110100110010110...
sage: derived_sequence(_, _[:1])
word: 0120210121020120210201210120210121020121...
sage: derived_sequence(_, _[:1])
word: 0123013201232013012301320130123201230132...
sage: derived_sequence(_, _[:1])
word: 0123013201232013012301320130123201230132...


Note

Note that method return_words_derivate of finite words in Sage does the same for finite words but without returning the translation dictionnary:

sage: w = words.ThueMorseWord()
sage: prefix = w[:1000]
sage: prefix.return_words_derivate(prefix[:1])
word: 1231321232131231321312321231321232131232...