Codings of rotations are full A. Blondin Mass e S. Brlek S. Labb - PowerPoint PPT Presentation

Codings of rotations are full A. Blondin Mass´ e S. Brlek S. Labb´ e L. Vuillon Universit´ e du Qu´ ebec ` a Montr´ eal EUROCOMB 2009 September 11th, 2009 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 1 / 15

Palindromes L L L R R R L ressasser tenet r reconocer d r r kisik d r e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 2 / 15

The Fibonacci word We define f − 1 = b , f 0 = a and, for n ≥ 1, f n = f n − 1 f n − 2 . Therefore, we have f 0 = a f 1 = ab f 2 = aba f 3 = abaab f 4 = abaababa f 5 = abaababaabaab . . . . . . The infinite word f ∞ is called the Fibonacci word. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 3 / 15

The Thue-Morse word We define t 0 = a and, for n ≥ 1, t n = t n − 1 t n − 1 . so that t 0 = a t 1 = ab t 2 = abba t 3 = abbabaab t 4 = abbabaabbaababba t 5 = abbabaabbaababbabaababbaabbabaab . . . . . . The infinite word t ∞ is called the Thue-Morse word. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 4 / 15

Number of distinct palindromic factors Theorem (Droubay, Justin and Pirillo, 2001) Let w be a finite word. Then | Pal ( w ) | ≤ | w | + 1 . w = p q Assume that the first occurrence of some palindromes p and q ends at the same position. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 5 / 15

Number of distinct palindromic factors Theorem (Droubay, Justin and Pirillo, 2001) Let w be a finite word. Then | Pal ( w ) | ≤ | w | + 1 . w = p q q Assume that the first occurrence of some palindromes p and q ends at the same position. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 5 / 15

Number of distinct palindromic factors Theorem (Droubay, Justin and Pirillo, 2001) Let w be a finite word. Then | Pal ( w ) | ≤ | w | + 1 . w = p q q Assume that the first occurrence of some palindromes p and q ends at the same position. Then p = q . e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 5 / 15

Number of distinct palindromic factors Theorem (Droubay, Justin and Pirillo, 2001) Let w be a finite word. Then | Pal ( w ) | ≤ | w | + 1 . w = p q q Assume that the first occurrence of some palindromes p and q ends at the same position. Then p = q . Theorem (Droubay, Justin and Pirillo, 2001) Sturmian words are full, i.e. they realize the upper bound. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 5 / 15

Palindromic complexity Number of palindrome factors Upper bound 500 Fibonacci word Thue-Morse word 400 300 200 100 Fixed point of a abb,b ba Length of prefix 100 200 300 400 500 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 6 / 15

The Fibonacci word is full w = a Palindromes a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b Palindromes a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a Palindromes a b a b a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a Palindromes a b a b a a a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a b Palindromes a b a b a a a b a a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a b a Palindromes a b a b a a a b a a b a b a a b a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a b a b Palindromes a b a b a a a b a a b a b a a b a b a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a b a b a Palindromes a b a b a a a b a a b a b a a b a b a b a b a b a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Fibonacci word is full w = a b a a b a b a a · · · Palindromes a b a b a a a b a a b a b a a b a b a b a b a b a a a b a b a a ... e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 7 / 15

The Thue-Morse word is lacunary w = a Palindromes a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b Palindromes a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b Palindromes a b b b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a Palindromes a b b b a b b a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a b Palindromes a b b b a b b a b a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a b a Palindromes a b b b a b b a b a b a b a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a b a a Palindromes a b b b a b b a b a b a b a a a e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a b a a b Palindromes a b b b a b b a b a b a b a a a b a a b e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

The Thue-Morse word is lacunary w = a b b a b a a b b · · · Palindromes a b b b a b b a b a b a b a a a b a a b − ... There is no new palindrome at this position! e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 8 / 15

Complete return words some occurrence next occurrence w = u u v We say that v is a complete return word of u in w , if v starts at an occurrence of u and ends at the end of the next occurrence of u . Fact A word w is full if and only if every complete return word of a palindrome factor of w is a palindrome. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 9 / 15

Codings of rotations (1/2) The coding of rotations of parameters ( x , α, β ) is the word 0 C = c 0 c 1 c 2 · · · such that � β 0 if x + i α ∈ [0 , β ) c i = x 1 if x + i α ∈ [ β, 1) 1 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 10 / 15

Codings of rotations (1/2) The coding of rotations of parameters ( x , α, β ) is the word 0 C = c 0 c 1 c 2 · · · such that � β 0 if x + i α ∈ [0 , β ) c i = x 1 if x + i α ∈ [ β, 1) x + α 11 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 10 / 15

Codings of rotations (1/2) The coding of rotations of parameters ( x , α, β ) is the word 0 C = c 0 c 1 c 2 · · · such that � β 0 if x + i α ∈ [0 , β ) c i = x 1 if x + i α ∈ [ β, 1) x + 2 α x + α 111 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 10 / 15

Codings of rotations (1/2) The coding of rotations of parameters ( x , α, β ) is the word 0 C = c 0 c 1 c 2 · · · such that � β 0 if x + i α ∈ [0 , β ) x + 3 α c i = x 1 if x + i α ∈ [ β, 1) x + 2 α x + α 1111 e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 10 / 15

Codings of rotations (1/2) The coding of rotations of parameters ( x , α, β ) is the word 0 C = c 0 c 1 c 2 · · · such that � β 0 if x + i α ∈ [0 , β ) x + 3 α c i = x 1 if x + i α ∈ [ β, 1) x + 2 α x + α 11111000000111 · · · e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 10 / 15

Codings of rotations (2/2) Many interesting problems related to codings of rotations: Density of the letters 0 and 1, Complexity, i.e. the number of factors of length n , or palindromic and f -palindromic complexity, Applications to number theory [Adamczewski, 2002], etc. In particular, Rote (1994) expressed sequences of complexity 2 n with respect to codings of rotations. e et al. (UQ` Blondin Mass´ AM, U. Savoie) Codings of rotations are full EUROCOMB 2009 11 / 15