Ecritures de nombres en base r eelle, fractals et pavages - PowerPoint PPT Presentation

´ Ecritures de nombres en base r´ eelle, fractals et pavages Wolfgang Steiner LIAFA, CNRS, Universit´ e Paris 7 26 octobre 2009 RAIM’09, Lyon

Digital expansions in base β Let β ≥ 2 be an integer. The β -expansion (binary, ternary, decimal, . . . ) of x ∈ [0 , 1) is given by the β -transformation T β : [0 , 1) → [0 , 1) , x �→ T β ( x ) = β x − ⌊ β x ⌋ , β = 2 β = 10 where ⌊ y ⌋ = max { n ∈ Z | n ≤ y } .

Digital expansions in base β Let β ≥ 2 be an integer. The β -expansion (binary, ternary, decimal, . . . ) of x ∈ [0 , 1) is given by the β -transformation T β : [0 , 1) → [0 , 1) , x �→ T β ( x ) = β x − ⌊ β x ⌋ , β = 2 β = 10 where ⌊ y ⌋ = max { n ∈ Z | n ≤ y } . We have x = ⌊ β x ⌋ + T β ( x ) β β

Digital expansions in base β Let β ≥ 2 be an integer. The β -expansion (binary, ternary, decimal, . . . ) of x ∈ [0 , 1) is given by the β -transformation T β : [0 , 1) → [0 , 1) , x �→ T β ( x ) = β x − ⌊ β x ⌋ , β = 2 β = 10 where ⌊ y ⌋ = max { n ∈ Z | n ≤ y } . We have T 2 β ( x ) x = ⌊ β x ⌋ + T β ( x ) = ⌊ β x ⌋ + ⌊ β T β ( x ) ⌋ + β 2 β 2 β β β

Digital expansions in base β Let β ≥ 2 be an integer. The β -expansion (binary, ternary, decimal, . . . ) of x ∈ [0 , 1) is given by the β -transformation T β : [0 , 1) → [0 , 1) , x �→ T β ( x ) = β x − ⌊ β x ⌋ , β = 2 β = 10 where ⌊ y ⌋ = max { n ∈ Z | n ≤ y } . We have T 2 � ∞ β ( x ) x = ⌊ β x ⌋ + T β ( x ) = ⌊ β x ⌋ + ⌊ β T β ( x ) ⌋ b n + = β 2 β 2 β n β β β n =1 with b n = ⌊ β T n − 1 ( x ) ⌋ ∈ { 0 , 1 , . . . , β − 1 } . Set d β ( x ) = b 1 b 2 · · · . β

Digital expansions in base β Let β > 1 be a real number. The (greedy) β -expansion of x ∈ [0 , 1) is given by the β -transformation T β : [0 , 1) → [0 , 1) , x �→ T β ( x ) = β x − ⌊ β x ⌋ , √ β = (1 + 5) / 2 β = 2 β = 10 where ⌊ y ⌋ = max { n ∈ Z | n ≤ y } . We have T 2 � ∞ β ( x ) + ⌊ β T β ( x ) ⌋ x = ⌊ β x ⌋ + T β ( x ) = ⌊ β x ⌋ b n + = β 2 β 2 β n β β β n =1 with b n = ⌊ β T n − 1 ( x ) ⌋ ∈ { 0 , 1 , . . . , ⌈ β ⌉ − 1 } . Set d β ( x ) = b 1 b 2 · · · . β

Admissible sequences The infinite expansion of 1 in base β is 1 = � ∞ n =1 a n β − n , where � � β � T n − 1 − 1 is given by the transformation a n = (1) β � � T β : (0 , 1] → (0 , 1] , x �→ � � T β ( x ) = β x − ⌈ β x ⌉ − 1 and ⌈ y ⌉ = min { n ∈ Z | n ≥ y } . Theorem (Parry 1960) We have b 1 b 2 · · · = d β ( x ) for some x ∈ [0 , 1) if and only if b n ∈ N and b n b n +1 · · · < lex a 1 a 2 · · · ∀ n ≥ 1 . Such a sequence b 1 b 2 · · · is called β -admissible. Examples β ∈ N : a 1 a 2 · · · = ( β − 1) ω , every sequence in { 0 , 1 , . . . , β − 1 } ω not terminating by ( β − 1) ω is β -admissible √ 5) / 2: a 1 a 2 · · · = (1 0) ω , every sequence in { 0 , 1 } ω without 1 1 β = (1 + and not terminating by (1 0) ω is β -admissible

Periodic β -expansions for Pisot numbers β Pisot number: algebraic integer β > 1 with | α | < 1 for every Galois conjugate α � = β ; in particular every integer β ≥ 2 If β ≥ 2 is an integer, then d β ( x ) is eventually periodic if and only if x ∈ Q ∩ [0 , 1), d β ( x ) is purely periodic if and only if the denominator of x is coprime with β .

Periodic β -expansions for Pisot numbers β Pisot number: algebraic integer β > 1 with | α | < 1 for every Galois conjugate α � = β ; in particular every integer β ≥ 2 If β ≥ 2 is an integer, then d β ( x ) is eventually periodic if and only if x ∈ Q ∩ [0 , 1), d β ( x ) is purely periodic if and only if the denominator of x is coprime with β . Theorem (Schmidt 1980) If β is Pisot, d β ( x ) is eventually periodic iff x ∈ Q ( β ) ∩ [0 , 1) . If d β ( x ) is eventually periodic for every x ∈ Q ∩ [0 , 1) , then β is Pisot or Salem ( | α | ≤ 1 for every Galois conjugate α � = β ).

Periodic β -expansions for Pisot numbers β Pisot number: algebraic integer β > 1 with | α | < 1 for every Galois conjugate α � = β ; in particular every integer β ≥ 2 If β ≥ 2 is an integer, then d β ( x ) is eventually periodic if and only if x ∈ Q ∩ [0 , 1), d β ( x ) is purely periodic if and only if the denominator of x is coprime with β . Theorem (Schmidt 1980) If β is Pisot, d β ( x ) is eventually periodic iff x ∈ Q ( β ) ∩ [0 , 1) . If d β ( x ) is eventually periodic for every x ∈ Q ∩ [0 , 1) , then β is Pisot or Salem ( | α | ≤ 1 for every Galois conjugate α � = β ). If β 2 − n β − 1 = 0 for some n ∈ Z , n ≥ 1 , then d β ( x ) is purely periodic for every x ∈ Q ∩ [0 , 1) . Lemma (Akiyama 1998) If β has a positive Galois conjugate (in particular if β 2 − n β + 1 = 0 ), then d β ( x ) is not purely periodic for any x ∈ Q ∩ (0 , 1) .

Natural extension of T β for Pisot units β Let β be a Pisot number, M β a companion matrix to the minimal polynomial X d − c 1 X d − 1 − c 2 X d − 2 − · · · − c d ∈ Z [ X ] of β ,   · · · · · · c 1 c 2 c d   1 0 · · · · · · 0     . ... ... .   M β = 0 . .    . .  ... ... ... . .   . . · · · 0 0 1 0 M β is expanding by the factor β on E β = R ( β d − 1 , . . . , β, 1) t , contracting on a hyperplane H of R d (spanned by the eigenvectors corresponding to the conjugates of β ). Let π be the projection on E β along H and e 1 = (1 , 0 , . . . , 0) t = e β − e H with e β = π ( e 1 ) ∈ E β , e H ∈ H .

Natural extension of T β for Pisot units β Let e 1 = e β − e H , � � S β = ( b n ) n ∈ Z | b n b n +1 · · · is β -admissible ∀ n ∈ Z , 0 � ∞ � b n M − n ψ : S β → R d , ( b n ) n ∈ Z �→ b n β − n e β + , β e H n = −∞ n =1 � �� � � �� � ∈ [0 , 1) ∈ H � T β : X β = ψ ( S β ) → X β , x �→ M β x − b 1 e 1 . For x = x e β + y , x ∈ [0 , 1), y ∈ H , we have � T β ( x e β + y ) = ( β x − b 1 ) e β + M β y + b 1 e H , � �� � T β ( x ) thus � T β ◦ ψ = ψ ◦ σ , where σ is the left-shift, and π ◦ � T β = T β ◦ π . If β is a Pisot unit ( | det M β | = | c d | = 1), then � T β is bijective except on a set of measure 0, ( � T β , X β ) is a natural extension of ( T β , [0 , 1)). T β is a toral automorphism since � � T β ( x ) ≡ M β x (mod Z d ).

Natural extensions for quadratic Pisot units β e 2 e 2 � T β ( X β, 1 ) β 2 = β + 1 e H e β e H e β β ≈ 1 . 618 X β, 0 X β, 1 � T β ( X β, 0 ) (golden mean) e 1 e 1 � T β ( X β, 2 ) e 2 e 2 β 2 = 3 β − 1 � T β ( X β, 1 ) X β, 1 X β, 2 β ≈ 2 . 618 X β, 0 e H e H (square of the e β � e β T β ( X β, 0 ) golden mean) e 1 e 1 �� �� , � X β, k = ψ ( b n ) n ∈ Z ∈ S β | b 1 = k T ( X β, k ) = M β X β, k − k e 1

Natural extensions for cubic Pisot units β β 3 = β 2 + β + 1, β ≈ 1 . 8393 β 3 = β + 1, β ≈ 1 . 3247 (Tribonacci number) (smallest Pisot number) e 2 e 2 e β e β e 1 e 3 e 1 e 3

Shape of X β Since X β = ψ ( S β ) with � � S β = ( b n ) n ∈ Z | b n b n +1 · · · is β -admissible ∀ n ∈ Z , 0 � ∞ � ψ : S β → R d , ( b n ) n ∈ Z �→ b n β − n e β + b n M − n β e H , n =1 n = −∞ we have � � � X β = x e β + D β ( x ) , x ∈ [0 , 1) where � � � 0 � � � b n M − n D β ( x ) = � ( b n ) n ∈ Z ∈ S β , b 1 b 2 · · · = d β ( x ) β e H . n = −∞ Lemma �� � T n If β is a Pisot number, then V β = β (1) | n ≥ 0 is a finite set. We have D β ( x ) ⊇ D β ( y ) if 0 ≤ x ≤ y < 1 , with D β ( x ) = D β ( y ) if � � and only if [ x , y ) ∩ V β = ∅ , hence # D β ( x ) | x ∈ [0 , 1) = # V β . D β ( x ) is compact for every x ∈ [0 , 1) .

Determining digits in d β ( x ) Let β be a Pisot unit, x ∈ [0 , 1) and d β ( x ) = b 1 b 2 · · · . We have b n = k if and only if � T n − 1 ( x e β ) ∈ X β, k . Note that β x β n − 1 e β = M n − 1 ( x e β ) ≡ � T n − 1 (mod Z d ) , ( x e β ) β β thus x β n − 1 e β ∈ X β, k (mod Z d ) if b n = k . Conjecture If β is a Pisot unit, then the intersection of X β, k and X β,ℓ mod Z d has Lebesgue measure zero for every ℓ � = k.