Crystallographic number systems Beno t Loridant University of - PowerPoint PPT Presentation

Crystallographic number systems Benoˆ ıt Loridant University of Leoben, Austria Prag - May, 2008 Supported by FWF, Project S9610.

Introduction. Purpose: dynamical systems associated to fractal tiles providing crystallographic tilings.

Introduction. Purpose: dynamical systems associated to fractal tiles providing crystallographic tilings. Questions: attractor of the dynamical systems, topology of the tiles.

Introduction. Purpose: dynamical systems associated to fractal tiles providing crystallographic tilings. Questions: attractor of the dynamical systems, topology of the tiles. Results: correspondances with canonical number systems.

Crystallographic groups. Γ ≤ Isom ( R n ) is a crystallographic group if Γ ≃ Z n ⋉ { id , r 2 , . . . , r d } , where r 2 , . . . , r d are isometries of finite order.

Crystallographic groups. Γ ≤ Isom ( R n ) is a crystallographic group if Γ ≃ Z n ⋉ { id , r 2 , . . . , r d } , where r 2 , . . . , r d are isometries of finite order. n = 2: 17 crystallographic groups. Example : a ( x , y ) = ( x +1 , y ) , b ( x , y ) = ( x , y +1) , c ( x , y ) = ( − x , − y ) . A p 2-group is isomorphic to the group generated by a , b , c .

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group,

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group, g expanding affine mapping such that g Γ g − 1 ≤ Γ,

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group, g expanding affine mapping such that g Γ g − 1 ≤ Γ, D ⊂ Γ finite complete set of right coset representatives of Γ / g Γ g − 1 : Γ = � δ ∈D g Γ g − 1 δ (disjoint).

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group, g expanding affine mapping such that g Γ g − 1 ≤ Γ, D ⊂ Γ finite complete set of right coset representatives of Γ / g Γ g − 1 : Γ = � δ ∈D g Γ g − 1 δ (disjoint). A crystile with respect to (Γ , g , D ) is a compact set T = T o ⊂ R n , such that

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group, g expanding affine mapping such that g Γ g − 1 ≤ Γ, D ⊂ Γ finite complete set of right coset representatives of Γ / g Γ g − 1 : Γ = � δ ∈D g Γ g − 1 δ (disjoint). A crystile with respect to (Γ , g , D ) is a compact set T = T o ⊂ R n , such that � R n = γ ( T ) (1) γ ∈ Γ without overlapping ( tiling property )

Crystallographic reptiles (Gelbrich, 1994). Let Γ crystallographic group, g expanding affine mapping such that g Γ g − 1 ≤ Γ, D ⊂ Γ finite complete set of right coset representatives of Γ / g Γ g − 1 : Γ = � δ ∈D g Γ g − 1 δ (disjoint). A crystile with respect to (Γ , g , D ) is a compact set T = T o ⊂ R n , such that � R n = γ ( T ) (1) γ ∈ Γ without overlapping ( tiling property ) and � g ( T ) = δ ( T ) (2) δ ∈D ( replication property ).

Example of a p 2 -crystile. � 0 � 1 � � x � � − 3 g ( x , y ) = + . 1 − 1 0 y

Example of a p 2 -crystile. � 0 � 1 � � x � � − 3 g ( x , y ) = + . 1 − 1 0 y Figure: T defined by g ( T ) = T ∪ a ( T ) ∪ c ( T ) and its neighbors.

Associated dynamical system. (Γ , g , D ) crystile data with id ∈ D . Γ = g Γ g − 1 D .

Associated dynamical system. (Γ , g , D ) crystile data with id ∈ D . Γ = g Γ g − 1 D . Define Φ : Γ → Γ Φ( γ ) such that γ = g Φ( γ ) g − 1 δ. γ �→

Associated dynamical system. (Γ , g , D ) crystile data with id ∈ D . Γ = g Γ g − 1 D . Define Φ : Γ → Γ Φ( γ ) such that γ = g Φ( γ ) g − 1 δ. γ �→ δ ∈ D and Φ( γ ) are uniquely defined by γ .

Associated dynamical system. (Γ , g , D ) crystile data with id ∈ D . Γ = g Γ g − 1 D . Define Φ : Γ → Γ Φ( γ ) such that γ = g Φ( γ ) g − 1 δ. γ �→ δ ∈ D and Φ( γ ) are uniquely defined by γ . Iterating Φ, one gets g γ 1 g − 1 δ 0 γ = g g γ 2 g − 1 δ 1 g − 1 δ 0 = = . . . g m Φ m ( γ ) g − 1 δ m − 1 . . . g − 1 δ 1 g − 1 δ 0 = with digits δ 0 , . . . , δ m − 1 ∈ D .

Crystallographic number system. Definition. (Γ , g , D ) is a crystallographic number system if for every γ ∈ Γ, Φ m ( γ ) = id for some m ∈ N . One then writes γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g or just ( δ m − 1 , . . . , δ 0 ) g .

Crystallographic number system. Definition. (Γ , g , D ) is a crystallographic number system if for every γ ∈ Γ, Φ m ( γ ) = id for some m ∈ N . One then writes γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g or just ( δ m − 1 , . . . , δ 0 ) g . Example. Consider Γ ≃ Z n and g ( x ) = M x with M expanding integer matrix.

Crystallographic number system. Definition. (Γ , g , D ) is a crystallographic number system if for every γ ∈ Γ, Φ m ( γ ) = id for some m ∈ N . One then writes γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g or just ( δ m − 1 , . . . , δ 0 ) g . Example. Consider Γ ≃ Z n and g ( x ) = M x with M expanding integer matrix. Then g Γ g − 1 ≤ Γ means M Z n ≤ Z n .

Crystallographic number system. Definition. (Γ , g , D ) is a crystallographic number system if for every γ ∈ Γ, Φ m ( γ ) = id for some m ∈ N . One then writes γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g or just ( δ m − 1 , . . . , δ 0 ) g . Example. Consider Γ ≃ Z n and g ( x ) = M x with M expanding integer matrix. Then g Γ g − 1 ≤ Γ means M Z n ≤ Z n . The digit set has the form � p i � � � D = x �→ x + ; 1 ≤ i ≤ d . q i

Crystallographic number system. Definition. (Γ , g , D ) is a crystallographic number system if for every γ ∈ Γ, Φ m ( γ ) = id for some m ∈ N . One then writes γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g or just ( δ m − 1 , . . . , δ 0 ) g . Example. Consider Γ ≃ Z n and g ( x ) = M x with M expanding integer matrix. Then g Γ g − 1 ≤ Γ means M Z n ≤ Z n . The digit set has the form � p i � � � D = x �→ x + ; 1 ≤ i ≤ d . q i (Γ , g , D ) is a crystallographic number system iff �� p i � � �� M , N := ; 1 ≤ i ≤ d is a number system . q i

Characterization of Crystems : counting automaton. (Γ , g , D ) given.

Characterization of Crystems : counting automaton. (Γ , g , D ) given. States: γ ∈ Γ. δ | δ ′ → γ ′ iff δγ = g γ ′ g − 1 δ ′ . − − Edges: γ

Characterization of Crystems : counting automaton. (Γ , g , D ) given. States: γ ∈ Γ. δ | δ ′ → γ ′ iff δγ = g γ ′ g − 1 δ ′ . − − Edges: γ id | δ ′ Note that γ − − → Φ( γ ).

Characterization of Crystems : counting automaton. (Γ , g , D ) given. States: γ ∈ Γ. δ | δ ′ → γ ′ iff δγ = g γ ′ g − 1 δ ′ . − − Edges: γ id | δ ′ Note that γ − − → Φ( γ ). (Γ , g , D ) is a crystem iff for every γ there is a finite walk id | δ 0 id | δ 1 id | δ m − 1 − − − → γ 1 − − − → . . . − − − − → id γ in the counting automaton.

Characterization of Crystems : counting automaton. (Γ , g , D ) given. States: γ ∈ Γ. δ | δ ′ → γ ′ iff δγ = g γ ′ g − 1 δ ′ . − − Edges: γ id | δ ′ Note that γ − − → Φ( γ ). (Γ , g , D ) is a crystem iff for every γ there is a finite walk id | δ 0 id | δ 1 id | δ m − 1 − − − → γ 1 − − − → . . . − − − − → id γ in the counting automaton. If γ = ( id ∞ , δ m − 1 , . . . , δ 0 ) g , then γγ 0 = g m γ ′ g − m ( δ ′ m − 1 , . . . , δ ′ 0 ) g where δ m − 1 | δ ′ δ 0 | δ ′ δ 1 | δ ′ m − 1 → γ ′ . 0 1 γ 0 − − − → γ 1 − − − → . . . − − − − − −

Example of p 2-crystem (Γ = < a , b , c > ). � � x � � � � 0 1 0 g ( x , y ) = + . − 1 − 3 0 y 2

Example of p 2-crystem (Γ = < a , b , c > ). � � x � � � � 0 1 0 g ( x , y ) = + . − 1 − 3 0 y 2 Figure: T : g ( T ) = T ∪ b ( T ) ∪ c ( T ) and counting subautomaton.

Characterization by a subautomaton. (Γ , g , D ) is a crystem iff for every γ there is a finite walk id | δ 0 id | δ 1 id | δ m − 1 γ − − − → γ 1 − − − → . . . − − − − → id (3) in the counting automaton.

Characterization by a subautomaton. (Γ , g , D ) is a crystem iff for every γ there is a finite walk id | δ 0 id | δ 1 id | δ m − 1 γ − − − → γ 1 − − − → . . . − − − − → id (3) in the counting automaton. Suppose Property (3) is fulfilled by the states of a stable subautomaton that generates Γ. Then (Γ , g , D ) is a crystem.

Example of p 2-non-crystem (Γ = < a , b , c > ). � � x � � − 1 0 g ( x , y ) = . − 3 − 1 y

Example of p 2-non-crystem (Γ = < a , b , c > ). � � x � � − 1 0 g ( x , y ) = . − 3 − 1 y Figure: T : g ( T ) = T ∪ b ( T ) ∪ a − 1 c ( T ) and counting subautomaton.

Example of p 3-crystem √ Γ = < a , b , r > , b ( x , y ) = ( x + 1 2 ) , r = rot (0; 2 π 3 2 , y + 3 )). √ � � x � � 0 3 √ g ( x , y ) = . − 3 0 y