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A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle Fagnot 2 1 LIAFA, CNRS Universit e Paris 7 2 IGM, CNRS Universit e de Marne-la-Vall ee STACS2007, Aachen Olivier Carton (LIAFA)

  1. A First Investigation of Sturmian Trees Jean Berstel 2 , Luc Boasson 1 Olivier Carton 1 , Isabelle Fagnot 2 1 LIAFA, CNRS Universit´ e Paris 7 2 IGM, CNRS Universit´ e de Marne-la-Vall´ ee STACS’2007, Aachen Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 1 / 14

  2. Outline 1 Sturmian words 2 Sturmian trees 3 Slow automata 4 Rank and degree 5 Results Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 2 / 14

  3. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  4. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  5. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  6. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  7. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  8. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  9. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  10. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  11. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  12. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  13. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  14. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  15. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  16. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  17. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  18. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abab abaab Factors bab baab ababa baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  19. Factors of an infinite word A factor w of a word x is a finite word that occurs in x , that is, there are words u and y such that x = uwy . Example (Fibonacci word) x = a b a a b a b a a b a a b a b a a b a b a a b . . . Length: n 1 2 3 4 5 a aa aab aaba aabaa b ab aba abaa aabab ba baa abab abaab Factors bab baab ababa baba baba baaba babaa # 2 3 4 5 6 Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 3 / 14

  20. Sturmian words Proposition (Hedlund & Morse) An infinite word x ultimately periodic iff there is an integer n such that x has at most n distinct factors of length n . An infinite word x is Sturmian if the number of its factors of length n is n + 1 for each n . Sturmian words are non ultimately periodic words with the smallest complexity. Example (Fibonacci word: f n +2 = f n +1 f n ) f 0 = a f 1 = ab f 2 = aba f 3 = abaab f ω = abaababa · · · Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 4 / 14

  21. Characterization: cutting sequences y = βx + ρ x = a b b a b ab b a b ba b Theorem A infinite word is Sturmian iff it is the cutting sequence of a straight line y = βx + ρ with an irrational slope β . Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 5 / 14

  22. Factor of a tree A factor of height h of a tree t is a subtree of height h that occurs in t . • • • Factor • • • • of height 3 • • • • • • • • Factor of height 2 • • • • • • • • • • • • • • • • Olivier Carton (LIAFA) Sturmian Trees STACS’2007, Aachen 6 / 14

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