Extensions naturelles des. et pavages

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1 Extensions naturelles des bêta-transformations généralisées et pavages Wolfgang Steiner LIAFA, CNRS, Université Paris Diderot Paris 7 (travail en commun avec Charlene Kalle, Universiteit Utrecht, en ce moment à l University of Warwick 19 Janvier 2010 Séminaire de Probabilités et Théorie Ergodique, LAMFA, Amiens

2 Transformations generating digital expansions in base β Let β > 1 be a real number. Let A R be a finite set, the digit set. Let X = a A X a, where the X a s are bounded intervals (or finite unions of them and the union is disjoint, typically [ X a1 [ X a2 [ X a3 [ X a 4 Define the transformation T : X X by Assume that T (X = X. T (x = βx a if x X a.

3 Transformations generating digital expansions in base β Let β > 1 be a real number. Let A R be a finite set, the digit set. Let X = a A X a, where the X a s are bounded intervals (or finite unions of them and the union is disjoint, typically [ X a1 [ X a2 [ X a3 [ X a 4 Define the transformation T : X X by T (x = βx a if x X a. Assume that T (X = X. Define the digit sequence b(x = b 1 (xb 2 (x by Then x = b 1(x β + T (x β b n (x = a if T n 1 (x X a. = b 1(x β + b 2(x β 2 and we call b(x the T -expansion of x. + T 2 (x β 2 = = n=1 b n (x β n,

4 Example: 1 < β 2, A = {0, 1} βx 0 βx β 1 β 1

5 Example: 1 < β 2, A = {0, 1} (greedy β-transformation: X 0 = [0, 1/β X 1 = [1/β, 1 βx 0 βx 1 [ X 0 [ 0 1 β X β 1

6 Example: 1 < β 2, A = {0, 1} ( X 0 ]( X 1 ] (greedy β-transformation: X 0 = [0, 1/β X 1 = [1/β, 1 βx 0 βx 1 lazy β-transformation: X 0 = ( 2 β β 1, ] 1 β(β 1 X 1 = ( 1 β(β 1, ] 1 β 1 [ X 0 [ 0 1 β X β 1

7 Example: 1 < β 2, A = {0, 1} ( X 0 ]( X 1 ] (greedy β-transformation: X 0 = [0, 1/β X 1 = [1/β, 1 βx 0 βx 1 lazy β-transformation: X 0 = ( 2 β β 1, ] 1 β(β 1 X 1 = ( 1 β(β 1, ] 1 β 1 [ X 0 [ 0 1 β X β 1 intermediate transformations different alphabets

8 Natural extensions Consider the dynamical system (X, B, T, where B is the Lebesgue σ-algebra on X. The map T is not invertible. A natural extension of (X, B, T is an invertible system ( X, B, T such that: There is a surjective and measurable map π : X X with π T = T π. This system is the smallest in the sense of σ-algebras: T n (π 1 (B = B. n 0

9 Conditions on β, companion matrix, eigenvectors Let β > 1 be a Pisot unit, i.e., an algebraic integer with β j < 1 for every Galois conjugate β j β of β and minimal polynomial X d c 1 X d 1 c 2 X d 2 c d Z[X ] with c d { 1, 1}.

10 Conditions on β, companion matrix, eigenvectors Let β > 1 be a Pisot unit, i.e., an algebraic integer with β j < 1 for every Galois conjugate β j β of β and minimal polynomial X d c 1 X d 1 c 2 X d 2 c d Z[X ] with c d { 1, 1}. Let M be the companion matrix c 1 c 2 c d 1 c d M = , det M = c d =

11 Conditions on β, companion matrix, eigenvectors Let β > 1 be a Pisot unit, i.e., an algebraic integer with β j < 1 for every Galois conjugate β j β of β and minimal polynomial X d c 1 X d 1 c 2 X d 2 c d Z[X ] with c d { 1, 1}. Let M be the companion matrix c 1 c 2 c d 1 c d M = , det M = c d = For 1 j d, let v j = ν j (β d 1 j,..., β j, 1 t, ν j C, be a right eigenvector of M to the eigenvalue β j (with β 1 = β, such that d j=1 v j = e 1 = (1, 0,..., 0 t. Let H be the hyperplane of R d spanned by the real and imaginary parts of v 2,..., v d, then M is contractive on H. Let e 1 = e β + e H with e β = v 1 (Me β = β e β, e H = d j=2 v j H.

12 Natural extension domain Let S denote the set of two-sided sequences u = (u n n Z A Z such that every suffix of u is a T -expansion of some x X : S = {u A Z u n u n+1 b(x for all n Z}.

13 Natural extension domain Let S denote the set of two-sided sequences u = (u n n Z A Z such that every suffix of u is a T -expansion of some x X : S = {u A Z u n u n+1 b(x for all n Z}. Assume that A Z (for simplicity; with small modifications, everything holds for A Q(β. Define ψ : S R d by ψ(u = n 1 u n β n e β n 0 u n M n e H. Let X = ψ(s = a A X a be the natural extension domain, with X a = { ψ(u u S, u 1 = a }. Note that the union is disjoint since the e β -coordinates in X a are different from those in X a for all a a, and that X is bounded.

14 Natural extension transformation Define the natural extension transformation T : X X by T (x = Mx a e 1 for x X a.

15 Natural extension transformation Define the natural extension transformation T : X X by T (x = Mx a e 1 for x X a. Write x = x e β + y with x X, y H, then T (x = T (x e β + y = (βx a e }{{} β + My a e H. T (x Let π be the projection on the e β -coordinate, then π T = T π.

16 Natural extension transformation Define the natural extension transformation T : X X by T (x = Mx a e 1 for x X a. Write x = x e β + y with x X, y H, then T (x = T (x e β + y = (βx a e }{{} β + My a e H. T (x Let π be the projection on the e β -coordinate, then π T = T π. Since ψ(u = n 1 u n β n e β n 0 u n M n e H, we have T ( ψ(u = n 1 u n+1 β n e β n 0 u n+1 M n e H = ψ ( σ(u X, where σ denotes the left-shift on S, and T ( X = T ( ψ(s = ψ ( σ(s = ψ(s = X.

17 Natural extension Let λ d denote the Lebesgue measure on R d, then det M = 1 yields ( ( λ d T ( Xa = λd M Xa a e 1 = λd ( Xa. Recall that X = X a A a (disjoint and T ( X = X, hence ( λ d T ( Xa T ( Xa = 0 for a a. Therefore, T is bijective, up to a set of measure zero.

18 Natural extension Let λ d denote the Lebesgue measure on R d, then det M = 1 yields λ d ( T ( Xa = λd ( M Xa a e 1 = λd ( Xa. Recall that X = a A X a (disjoint and T ( X = X, hence λ d ( T ( Xa T ( Xa = 0 for a a. Therefore, T is bijective, up to a set of measure zero. It can be shown that Q d z Z d ( z + X, and that λ d ( X = λd ( closure ( X. Since X is bounded, 0 < λd ( X <.

19 Natural extension Let λ d denote the Lebesgue measure on R d, then det M = 1 yields λ d ( T ( Xa = λd ( M Xa a e 1 = λd ( Xa. Recall that X = a A X a (disjoint and T ( X = X, hence λ d ( T ( Xa T ( Xa = 0 for a a. Therefore, T is bijective, up to a set of measure zero. It can be shown that Q d z Z d ( z + X, and that λ d ( X = λd ( closure ( X. Since X is bounded, 0 < λd ( X <. Theorem ( X, B, T is a natural extension of (X, B, T, up to measure zero.

20 Natural extension Let λ d denote the Lebesgue measure on R d, then det M = 1 yields λ d ( T ( Xa = λd ( M Xa a e 1 = λd ( Xa. Recall that X = a A X a (disjoint and T ( X = X, hence λ d ( T ( Xa T ( Xa = 0 for a a. Therefore, T is bijective, up to a set of measure zero. It can be shown that Q d z Z d ( z + X, and that λ d ( X = λd ( closure ( X. Since X is bounded, 0 < λd ( X <. Theorem ( X, B, T is a natural extension of (X, B, T, up to measure zero. Since T preserves the Lebesgue measure, an ACIM of T is given by the projection of the Lebesgue measure of X on the e β -coordinate.

21 Examples of natural extensions for the β-transformation β 2 = β + 1 (β 1.618: X 0 X1 e β T ( X1 T ( X0 e β [ X 0 [ X1 e H e 1 e H e 1

22 Examples of natural extensions for the β-transformation β 2 = β + 1 (β 1.618: T ( X1 X 0 X1 e β T ( X0 e β [ X 0 [ X1 e H e 1 e H T ( X2 e 1 β 2 = 3β 1 (β 2.618: X 0 X1 X2 e β T ( X1 T ( X0 e β e 1 e 1 [ X0 [ X1 [ X2 e H e H

23 Examples of natural extensions for the β-transformation β 3 = β 2 + β + 1 (β β 3 = β + 1 (β (Tribonacci number (smallest Pisot number e β e β e 1 e 1

24 Natural extensions for minimal weight transformations If β is a Pisot unit and A Z, then let the weight of T be w(t = 1 λ d ( X a λ d ( Xa. a A T is a minimal weight transformation if w(t w(s for every S with the same β.

25 Natural extensions for minimal weight transformations If β is a Pisot unit and A Z, then let the weight of T be w(t = 1 λ d ( X a λ d ( Xa. a A T is a minimal weight transformation if w(t w(s for every S with the same β. Theorem (Frougny St 2008 Let T be defined by β > 1 and A = { 1, 0, 1}, 1 2 α 1 β 1, X 1 = [ α, α/β, X 0 = [ α/β, α/β, X 1 = [α/β, α. If β 2 = β + 1 and or β 3 = β 2 + β + 1 and β2 β 2 +1 α 2β β 2 +1, β β+1 α 2+1/β β+1, or β 3 = β + 1 and β3 β 2 +1 α β2 +1/β β 2 +1, then T is a minimal weight transformation.

26 Examples of minimal weight transformations β = , α = β2 +β 3 β X 1 e β X 1 X 0 e 1 e H λ 2 ( X0 = 4 5 T ( X 1 e β T ( X 0 e 1 T ( X 1 e H

27 Examples of minimal weight transformations β = , α = β2 +β 3 β β 3 = β 2 + β + 1, α = β β X 1 e β X 1 T ( X 1 X 0 e 1 e H λ 2 ( X0 = 4 5 e β β 3 = β + 1, α = β3 β T ( X 0 e 1 T ( X 1 e H

28 Structure of X Let D x = { y x e β + y X } be the x-fiber of X. D x is compact.

29 Structure of X Let D x = { y x e β + y X } be the x-fiber of X. D x is compact. Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k. Define X aj = (δ j 1, δ j ], X = a A X a, T : X X with T (x = βx a if x X a. δ 0 X a1 δ 1 X a2 δ 2 X a3 δ 3 X a4 δ 4 [ [ [ [ X a1 X a2 X a3 X a4 ( ]( ]( ]( ]

30 Structure of X Let D x = { y x e β + y X } be the x-fiber of X. D x is compact. Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k. Define X aj = (δ j 1, δ j ], X = a A X a, T : X X with T (x = βx a if x X a. δ 0 X a1 δ 1 X a2 δ 2 X a3 δ 3 X a4 δ 4 [ [ [ [ X a1 X a2 X a3 X a4 ( ]( ]( ]( ] Let m j > 0 be minimal such that T m j (δ j = T m j (δ j, 0 < j < k, with m j = if T n (δ j T n (δ j for all n > 0, and set V = {δ 0 } { T n (δ j, T n (δ j } \ {δ k }. 0<j<k 0<n<m j Proposition If x, y X, x < y and (x, y] V =, then D x = D y.

31 Structure of X Let D x = { y x e β + y X } be the x-fiber of X. D x is compact. Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k. Define X aj = (δ j 1, δ j ], X = a A X a, T : X X with T (x = βx a if x X a. δ 0 X a1 δ 1 X a2 δ 2 X a3 δ 3 X a4 δ 4 [ [ [ [ X a1 X a2 X a3 X a4 ( ]( ]( ]( ] Let m j > 0 be minimal such that T m j (δ j = T m j (δ j, 0 < j < k, with m j = if T n (δ j T n (δ j for all n > 0, and set V = {δ 0 } { T n (δ j, T n (δ j } \ {δ k }. 0<j<k 0<n<m j Proposition If x, y X, x < y and (x, y] V =, then D x = D y. V is finite if and only if m j < or the T - and T -orbits of δ j are eventually periodic, 0 < j < k.

32 Periodic T -expansions Theorem The T -expansion (and T -expansion of x is eventually periodic if and only if x Q(β X. (Here, β can be a Pisot number, not necessarily a Pisot unit. cf. Bertrand (1977, K. Schmidt (1980, Frank Robinson (2008

33 Periodic T -expansions Theorem The T -expansion (and T -expansion of x is eventually periodic if and only if x Q(β X. (Here, β can be a Pisot number, not necessarily a Pisot unit. cf. Bertrand (1977, K. Schmidt (1980, Frank Robinson (2008 For x Q(β, let x (j be the image of x by the automorphism from Q(β to Q(β j mapping β to β j, and Ψ : Q(β Q d, in particular Ψ(x = x e 1 if x Q. x d x (j v j, j=1 Theorem The T -expansion of x is purely periodic if and only if x Q(β X and Ψ(x X. cf. Ito Rao (2006, Berthé Siegel (2005

34 (Multiple Tilings of H Let Φ : Q(β H, x For x Z[β] X, set d x (j v j = Ψ(x xe β, T x = Lim Φ( β n T n (x = Lim n n Mn Φ ( T n (x = Φ(x D x, where Lim denotes the Hausdorff limit. j=2

35 (Multiple Tilings of H Let Φ : Q(β H, x For x Z[β] X, set d x (j v j = Ψ(x xe β, T x = Lim Φ( β n T n (x = Lim n n Mn Φ ( T n (x = Φ(x D x, where Lim denotes the Hausdorff limit. Every tile T x subdivides into contracted copies of other tiles: T x = M T y, D x = ( M D y Φ ( b 1 (y. y T 1 (x j=2 y T 1 (x If V is finite, this gives a graph-directed iterated function system for {D x } x V, the unions are disjoint up to sets of measure zero, λ d 1 ( T x = 0; λ d 1 (T x > 0 iff x is in the support of the ACIM.

36 (Multiple Tilings of H Let Φ : Q(β H, x For x Z[β] X, set d x (j v j = Ψ(x xe β, T x = Lim Φ( β n T n (x = Lim n n Mn Φ ( T n (x = Φ(x D x, where Lim denotes the Hausdorff limit. Every tile T x subdivides into contracted copies of other tiles: T x = M T y, D x = ( M D y Φ ( b 1 (y. y T 1 (x j=2 y T 1 (x If V is finite, this gives a graph-directed iterated function system for {D x } x V, the unions are disjoint up to sets of measure zero, λ d 1 ( T x = 0; λ d 1 (T x > 0 iff x is in the support of the ACIM. Theorem If V is finite, then {T x } x Z[β] X is a multiple tiling of H. (There exists m 1 such that almost every point lies in exactly m tiles. cf. Thurston (1989, Praggastis (1999, Akiyama (1999, 2002, Ito Rao (2006, Berthé Siegel (2005

37 Examples of tilings of H for the β-transformations β 3 = β 2 + β + 1 β 3 = β + 1 T 3β2 5β T 3β2 +4β T 4β2 +3β+4 T 3β2 4β 2 T 2β2 +4β 1 T 3β2 +3β+2 T 4β2 +2β+5 T 3β2 3β 4 T 2β2 4β+1 T 2β2 +3β T 3β2 +2β+3 T 2β2 β 4 T 2β2 2β 3 T β2 β 1 T 2β2 3β 1 T β2 2β+1 T β2 3β+3 T 2β+4 T β2 +3β 2 T3β 3 T2β 2 T 2β2 +2β+1 T β2 +2β T β2 +β+1 T 3β2 +β+4 T 2β2 +β+3 T 2β2 +4 T β2 3 T β+2 Tβ 1 T β2 +2 T β2 β+4 T0 T β2 +β 3 T0 T2β 3 T β2 +3β 2 Tβ 1 T β2 +2β T β2 +β+2 T 2β2 +2β+4 T β2 +4 T 2β2 +β+5 T 2β2 +β 4 T 3β2 5 T 2β2 3 T β2 1 T 2β2 β 2 T β2 β T β+2 T β2 2β+1 T 2β+3 T 2β2 2β T β2 +4β 3 T 2β2 +3β+2 T 3β2 β 3 T 2β2 +4β T 3β2 +3β+5 T 4β2 β 5 T 3β2 2β 2 T 2β2 3β+1 T 2β2 +5β 2 T 3β2 +4β+3 T 4β2 2β 4 T 3β2 3β 1 T 3β2 +5β+1 T 5β2 2β 6 T 4β2 3β 3 Conjecture Let β be a Pisot unit and T : [0, 1 [0, 1, x βx βx, be the (greedy β-transformation. Then {T x } x Z[β] X is a tiling of H. T 3β2 4β+1

38 Sofic shifts Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k, and denote the T -expansion of x X by b(x. Theorem A sequence a v1 a v2 is the T -expansion of some x X iff b(δ vn 1 lex a vn a vn+1 < lex b(δ vn for all n 1.

39 Sofic shifts Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k, and denote the T -expansion of x X by b(x. Theorem A sequence a v1 a v2 is the T -expansion of some x X iff The closure of b(δ vn 1 lex a vn a vn+1 < lex b(δ vn for all n 1. S = {u A Z u n u n+1 b(x for all n Z} is a sofic shift if and only if b(δ j 1 and b(δ j are eventually periodic for all 1 j k, i.e., δ j Q(β.

40 Sofic shifts Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k, and denote the T -expansion of x X by b(x. Theorem A sequence a v1 a v2 is the T -expansion of some x X iff The closure of b(δ vn 1 lex a vn a vn+1 < lex b(δ vn for all n 1. S = {u A Z u n u n+1 b(x for all n Z} is a sofic shift if and only if b(δ j 1 and b(δ j are eventually periodic for all 1 j k, i.e., δ j Q(β. {T x } x Z[β] X can be a tiling even if the closure of S is not sofic!

41 Tiling of R d by X, partition of R d /Z d by { Xa }a A Theorem {T x } x Z[β] X is a tiling of H (multiple tiling of degree m = 1 if and only if {z + X } z Z d is a tiling of R d. cf. Ito Rao (2006

42 Tiling of R d by X, partition of R d /Z d by { Xa }a A Theorem {T x } x Z[β] X is a tiling of H (multiple tiling of degree m = 1 if and only if {z + X } z Z d is a tiling of R d. cf. Ito Rao (2006 Since T (x = Mx a e 1 if x X a, we have T (x M x (mod Z d.

43 Tiling of R d by X, partition of R d /Z d by { Xa }a A Theorem {T x } x Z[β] X is a tiling of H (multiple tiling of degree m = 1 if and only if {z + X } z Z d is a tiling of R d. cf. Ito Rao (2006 Since T (x = Mx a e 1 if x X a, we have T (x M x (mod Z d. Theorem If 0 D x, then β n 1 x e β X bn(x (mod Z d. If {z + X } z Z d is a tiling of R d and 0 is an inner point of D x, then b n (x = a if and only if β n 1 x e β X a (mod Z d. (This is also true for most x if 0 is not an inner point of D x.

44 Tiling of R d by X, partition of R d /Z d by { Xa }a A Theorem {T x } x Z[β] X is a tiling of H (multiple tiling of degree m = 1 if and only if {z + X } z Z d is a tiling of R d. cf. Ito Rao (2006 Since T (x = Mx a e 1 if x X a, we have T (x M x (mod Z d. Theorem If 0 D x, then β n 1 x e β X bn(x (mod Z d. If {z + X } z Z d is a tiling of R d and 0 is an inner point of D x, then b n (x = a if and only if β n 1 x e β X a (mod Z d. (This is also true for most x if 0 is not an inner point of D x. This can be used e.g. to calculate the statistics of the n-th digit in {P(m m Z, P(m X } for a polynomial P, cf. St (2002.

45 Degree of the multiple tiling The set P = {x Z[β] X b(x is purely periodic} is finite.

46 Degree of the multiple tiling The set P = {x Z[β] X b(x is purely periodic} is finite. Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k, and set ε = β min{δ j x 1 j k, x P X aj } > 0. Proposition Let z Z[β] [0, and n 0 such that β n z [0, ε. Then Φ(z lies exactly in the tiles T T n (x+β n z, x P. Φ ( Z[β] [0, is dense in H.

47 Degree of the multiple tiling The set P = {x Z[β] X b(x is purely periodic} is finite. Assume A = {a 1,..., a k }, X aj = [δ j 1, δ j, δ 0 < δ 1 < < δ k, and set ε = β min{δ j x 1 j k, x P X aj } > 0. Proposition Let z Z[β] [0, and n 0 such that β n z [0, ε. Then Φ(z lies exactly in the tiles T T n (x+β n z, x P. Φ ( Z[β] [0, is dense in H. Consider two properties: (F : P consists only of one element. (W : y P : x P z Z[β] [0, ε, n 0 : T n (x + z = T n (y + z = y. Theorem {T x } x Z[β] X is a tiling (m = 1 if and only if (W holds. cf. Akiyama (2002; (F (W

48 Example of a multiple tiling β = A = { 1, 1} X 1 = [ 1, 0, X 1 = [0, 1 multiple tiling of degree 4 X 1 e β e X1 1 e H T ( X1 e β [ X 1 [ X 1 e 1 T ( X 1 e H

49 Symmetric β-transformations (Akiyama Scheicher 2007 X = [ 1 2, 1 2, T (x = βx βx β 3: A = { 1, 0, 1}, X 1 = [ 1 2, 1 2β, X0 = [ 1 2β, 1 2β Example: β = tiling, X1 = [ 1 2β, 1 2 X 1 X0 X1 e β T ( X 1 X 0 e β [X 1 [ X 0 [ X1 v 2 T ( X 1 v 2

50 T.00 10(10 1 ω T ( 101 ω T.10(0 11 ω T.1 10(01 1 ω T.1 1(1 10 ω T.001( 110 ω T.0 1(1 10 ω T. 101( 110 ω T (10 1 ω T.10( 101 ω T.0(10 1 ω T.0(01 1 ω T.00( 101 ω symmetric β-transformation, β 3 = β 2 + β + 1 double tiling T.1( 110 ω T. 110( 101 ω T.( 101 ω T.(0 11 ω T.(1 10 ω T.( 110 ω T.(01 1 ω T.(10 1 ω T.1 10(10 1 ω T. 1(1 10 ω T.00(10 1 ω T.0(0 11 ω T.0( 101 ω T. 10(10 1 ω T.1 110( 101 ω T.10 1(1 10 ω T.01( 110 ω T.00 1(1 10 ω T. 11( 110 ω T. 110(0 11 ω T (10 1 ω T. 10(01 1 ω T.0010( 101 ω

51 T ( 1010 ω T ( 1010 ω T (10 10 ω T (10 10 ω T ( 1010 ω T ( 1010ω T (10 10 ω T.1 1(10 10 ω T (10 10 ω T (10 10 ω symmetric β-transformation, β 3 = β tiling T. 11 1(10 10 ω T.01 1(10 10 ω T.1( 1010 T.(0 101 ω ω T.(10 10 ω T.( 1010 ω T.(010 1 T. 1(10 10 ω ω T.0 11( 1010 ω T.1 11( 1010 ω T ( 1010 ω T ( 1010 ω T. 11( 1010 ω T ( 1010 ω T (10 10 ω T (10 10 ω T (10 10 ω T ( 1010 ω T ( 1010 ω T (10 10ω

52 T (10 10 ω T.1 10(010 1 ω T (10 10 ω T (10 10 ω T.01( 1010 ω T. 11( 1010 ω T.0(0 101 ω T. 101( 1010 ω symmetric β-transformation, β 3 = 2β 2 β + 1 tiling T. 10(010 1 ω T.1( 1010 ω T.(0 101 ω T.(10 10 ω T.1 11( 1010 ω T. 1(10 10 ω T.(010 1 ω T.10(0 101 ω T. 11 1(10 10 ω T.( 1010 ω T.10 1(10 10 ω T.0(010 1 ω T.1 1(10 10 ω T.0 1(10 10 ω T.1 101( 1010 ω T ( 1010 ω T. 110(0 101 ω T ( 1010 ω

53 T ( ω T ( ω T ( ω T.10( ω T.1 1( ω symmetric β-transformation, β 3 = β + 1 double tiling T.1 110( T.( ω ω T.( ω T. 1010( T.10 10( ω ω T.( ω T.( T ( ω ω T. 11( ω T. 10( ω T.10 11( ω T ( ω T ( ω