-expansions and multiple tilings...-expansions and multiple tilings -expansions and multiple tilings...
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β-expansions and multiple tilings
β-expansions and multiple tilings
Charlene Kalle,joint work with Wolfgang Steiner
(PhD supervisor Karma Dajani)
29 October 2009
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Introduction
Let β > 1 and A = {a0, . . . , am} a set of real numbers witha0 < a1 < . . . < am. Expressions of the form
x =∞∑
n=1
bn
βn,
with bn ∈ A for all n ≥ 1, are called β-expansions with arbitrary digits.
This gives numbers in the interval[ a0
β − 1,
am
β − 1
].
β is called the base, A is the digit set and elements of A are calleddigits.
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β-expansions and multiple tilings > Introduction
Allowable digit sets
If, for a given β > 1, a set of real numbers A = {a0, . . . , am} satisfies
(i) a0 < . . . < am,
(ii) max1≤j≤m(aj − aj−1) ≤ am − a0
β − 1,
it is called an allowable digit set. Then
every x ∈[ a0
β − 1,
am
β − 1
]has a β-expansion with digits in A.
(Pedicini, 2005)
the minimal amount of digits in A is dβe.
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β-expansions and multiple tilings > Introduction
Allowable digit sets
If, for a given β > 1, a set of real numbers A = {a0, . . . , am} satisfies
(i) a0 < . . . < am,
(ii) max1≤j≤m(aj − aj−1) ≤ am − a0
β − 1,
it is called an allowable digit set. Then
every x ∈[ a0
β − 1,
am
β − 1
]has a β-expansion with digits in A.
(Pedicini, 2005)
the minimal amount of digits in A is dβe.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Allowable digit sets
If, for a given β > 1, a set of real numbers A = {a0, . . . , am} satisfies
(i) a0 < . . . < am,
(ii) max1≤j≤m(aj − aj−1) ≤ am − a0
β − 1,
it is called an allowable digit set. Then
every x ∈[ a0
β − 1,
am
β − 1
]has a β-expansion with digits in A.
(Pedicini, 2005)
the minimal amount of digits in A is dβe.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Outline
Introduce a class of transformations that generate β-expansions.
Characterize the set of digit sequences given by such atransformation.
For specific β’s (Pisot units) give a construction of a naturalextension for the transformation.
From the natural extension, get an absolutely continuous invariantmeasure.
Under a further assumption, construct a multiple tiling of aEuclidean space and give an example that shows that the Pisotconjecture does not hold in this more general setting.
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β-expansions and multiple tilings > Introduction
Outline
Introduce a class of transformations that generate β-expansions.
Characterize the set of digit sequences given by such atransformation.
For specific β’s (Pisot units) give a construction of a naturalextension for the transformation.
From the natural extension, get an absolutely continuous invariantmeasure.
Under a further assumption, construct a multiple tiling of aEuclidean space and give an example that shows that the Pisotconjecture does not hold in this more general setting.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Outline
Introduce a class of transformations that generate β-expansions.
Characterize the set of digit sequences given by such atransformation.
For specific β’s (Pisot units) give a construction of a naturalextension for the transformation.
From the natural extension, get an absolutely continuous invariantmeasure.
Under a further assumption, construct a multiple tiling of aEuclidean space and give an example that shows that the Pisotconjecture does not hold in this more general setting.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Outline
Introduce a class of transformations that generate β-expansions.
Characterize the set of digit sequences given by such atransformation.
For specific β’s (Pisot units) give a construction of a naturalextension for the transformation.
From the natural extension, get an absolutely continuous invariantmeasure.
Under a further assumption, construct a multiple tiling of aEuclidean space and give an example that shows that the Pisotconjecture does not hold in this more general setting.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Introduction
Outline
Introduce a class of transformations that generate β-expansions.
Characterize the set of digit sequences given by such atransformation.
For specific β’s (Pisot units) give a construction of a naturalextension for the transformation.
From the natural extension, get an absolutely continuous invariantmeasure.
Under a further assumption, construct a multiple tiling of aEuclidean space and give an example that shows that the Pisotconjecture does not hold in this more general setting.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Transformations and admissible sequences
Transformations
For each β > 1 and allowable digit set A = {a0, . . . , am} there existtransformations that generate β-expansions with digits in A byiteration.
Example: Classic β-expansions
Consider a non-integer β > 1 and digit set A = {0, 1, . . . , bβc}. This
gives ‘classic’ β-expansions for all x ∈[0,bβcβ − 1
]. A transformation
that generates these is
Tx =
{βx (mod 1), if x ∈ [0, 1),
βx − bβc, if x ∈[1, bβcβ−1
].
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic β-expansions
0 1β
2β
2β−1
1
βx
βx − 1
βx − 2
This is the classic greedyβ-transformation.
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic β-expansions
0 1β
2β
2β−1
2101
Assign a digit to each interval.Make a digit sequence by setting
b1(x) =
j , if x ∈
[jβ ,
j+1β−1
],
for 0 ≤ j ≤ bβc,bβc, if x ∈
[bβcβ ,
bβcβ−1
].
and bn(x) = b1(T n−1x) for n ≥ 1.Then we have Tx = βx − b1 andT 2x = βTx − b2, etc.
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic β-expansions
0 1β
x 2β
2β−1
2101
Assign a digit to each interval.Make a digit sequence by setting
b1(x) =
j , if x ∈
[jβ ,
j+1β−1
],
for 0 ≤ j ≤ bβc,bβc, if x ∈
[bβcβ ,
bβcβ−1
].
and bn(x) = b1(T n−1x) for n ≥ 1.Then we have Tx = βx − b1 andT 2x = βTx − b2, etc.
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic β-expansions
0 1βTx x 2
β2
β−1
2101
Assign a digit to each interval.Make a digit sequence by setting
b1(x) =
j , if x ∈
[jβ ,
j+1β−1
],
for 0 ≤ j ≤ bβc,bβc, if x ∈
[bβcβ ,
bβcβ−1
].
and bn(x) = b1(T n−1x) for n ≥ 1.Then we have Tx = βx − b1 andT 2x = βTx − b2, etc.
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic β-expansions
If Tx = βx − b1(x), then x = b1β + Tx
β . Iterating this, we get after nsteps,
x =b1
β+
b2
β2+
T 2x
β2= · · · =
b1
β+
b2
β2+ · · ·+ bn
βn+
T nx
βn.
Since T nx ∈[0, bβcβ−1
)for all n, this converges and gives
x =∞∑
n=1
bn
βn.
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β-expansions and multiple tilings > Transformations and admissible sequences
Other transformations: greedy and lazy
Take β to be the golden mean and A = {0, 1, 3}. These are the greedyand lazy β-transformations with digits in A. [Dajani & K., 2007]
0 1β
3β
3β−1
310
βx βx − 1
βx − 3
0 3 a 3β−1
310
βx
βx − 1
βx − 3
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β-expansions and multiple tilings > Transformations and admissible sequences
Other transformations: the minimal weight transformation
Take β to be the golden mean and A = {−1, 0, 1}. This is a minimalweight transformation, i.e. if an x has a finite β-expansion, then theexpansion generated by this transformation has the highest number of0’s. [Frougny & Steiner, 2009]
−β2
β2
−12
12
0βx + 1
βx
βx − 1
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β-expansions and multiple tilings > Transformations and admissible sequences
Other transformations: the linear mod 1 transformation
Take β > 1 and 0 ≤ α < 1. Suppose n < β + α ≤ n + 1. The linearmod 1 transformation below (Tx = βx + α (mod 1)) givesβ-expansions with digits in {j − α : 0 ≤ j ≤ n}.
01−αβ
2−αβ
1
α βx + α
βx − 1 + α
βx − 2 + α
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β-expansions and multiple tilings > Transformations and admissible sequences
The class of transformations
Given a real number β > 1 and a digit set A = {a0, . . . , am}, weconsider the class of transformations that have the followingproperties.
For each digit in the digit set ai , there is a bounded interval Xi
and if i 6= j , then Xi ∩ Xj = ∅.On the interval Xi the transformation is given by Tx = βx − ai .
If X =⋃
i :ai∈A Xi , then TX = X .
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β-expansions and multiple tilings > Transformations and admissible sequences
The class of transformations
Given a real number β > 1 and a digit set A = {a0, . . . , am}, weconsider the class of transformations that have the followingproperties.
For each digit in the digit set ai , there is a bounded interval Xi
and if i 6= j , then Xi ∩ Xj = ∅.On the interval Xi the transformation is given by Tx = βx − ai .
If X =⋃
i :ai∈A Xi , then TX = X .
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β-expansions and multiple tilings > Transformations and admissible sequences
The class of transformations
Given a real number β > 1 and a digit set A = {a0, . . . , am}, weconsider the class of transformations that have the followingproperties.
For each digit in the digit set ai , there is a bounded interval Xi
and if i 6= j , then Xi ∩ Xj = ∅.On the interval Xi the transformation is given by Tx = βx − ai .
If X =⋃
i :ai∈A Xi , then TX = X .
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β-expansions and multiple tilings > Transformations and admissible sequences
Admissible sequences: The golden mean
We can characterise the digit sequences generated by a transformation.For the classic greedy β-transformation with β the golden mean, wehave the following.
0 1β
1
101
0 1β
1
101
0 1β
1
101
0 can be followed by 0 or 1, but 1 is always followed by 0. This isgiven by the orbit of 1. Hence, T produces precisely the sequencesfrom the set {u1u2 · · · | unun+1 6= 11, n ≥ 1}.
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β-expansions and multiple tilings > Transformations and admissible sequences
The set of admissible sequences
Given a transformation T for a β > 1 and digit set A, we call asequence u1u2 · · · ∈ AN admissible for T if there is an x ∈ X such thatu1u2 · · · = b(x).
A two-sided sequence · · · u−1u0u1 · · · is called admissible if for eachn ∈ Z there is an x ∈ X , such that unun+1 · · · = b(x).
Notation: S is the set of two sided admissible sequences.
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β-expansions and multiple tilings > Transformations and admissible sequences
Relation between expansions and sequences
To a β-transformation on an interval, there corresponds ashift-transformation on a set of digit sequences.
T
x =∑∞
n=1bnβn ⇒ b(x) = b1b2 · · · .
Tx =∑∞
n=1bn+1
βn ⇒ b(Tx) = b2b3 · · · .
T is not invertible: after applying T to x we ‘lose’ the first digit b1.
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β-expansions and multiple tilings > Transformations and admissible sequences
Admissible sequences
We can characterise the digit sequences generated by a transformation.
γ0 γ1
(a)γ2 γ3
a2a1a0
γ0 γ1
(b)γ2 γ3
a2a1a0
Let b(x) be a digit sequence given by (a) and b(x) the one given by(b). Then we have the following characterization in terms of thesequences b(γj) and b(γj).
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β-expansions and multiple tilings > Transformations and admissible sequences
Admissible sequences
γ0 γ1 γ2 γ3
a2a1a0
T :
γ0 γ1 γ2 γ3
a2a1a0
T :
Admissible sequences
A sequence u1u2 · · · ∈ {a0, . . . , am}N is generated by T iff for eachn ≥ 1, if un = aj , then
b(γj) � unun+1 · · · ≺ b(γj+1),
where � denotes the lexicographical ordering.
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β-expansions and multiple tilings > Transformations and admissible sequences
The classic admissible sequences
0 1β
2β
1
In the case of the classic greedy β-transformation, only the orbit of 1is important. This gives the Parrycondition.
Theorem (Parry, 1960)
Let b(1) be the expansion of 1generated by T . Then a sequenceu1u2 · · · ∈ {0, 1, . . . , bβc}N isgenerated by T iff for each n ≥ 1,
unun+1 · · · ≺ b(1).
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β-expansions and multiple tilings > Natural extensions and invariant measures
Invariant measures
The classic greedy β-transformation Tβ has the following properties.
It has an invariant measure that is equivalent to the Lebesguemeasure on the unit interval [0, 1). (Renyi, 1957)
The density function is given by
hc : [0, 1)→ [0, 1) : x 7→ 1
F (β)
∞∑n=0
1
βn1[0,T n1)(x),
where F (β) is a normalizing constant. (Gel’fond, 1959, and Parry,1960)
In general, one knows that an invariant measure equivalent to theLebesgue measure exists for these transformations (Lasota and Yorke,1974).
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β-expansions and multiple tilings > Natural extensions and invariant measures
Invariant measures
The classic greedy β-transformation Tβ has the following properties.
It has an invariant measure that is equivalent to the Lebesguemeasure on the unit interval [0, 1). (Renyi, 1957)
The density function is given by
hc : [0, 1)→ [0, 1) : x 7→ 1
F (β)
∞∑n=0
1
βn1[0,T n1)(x),
where F (β) is a normalizing constant. (Gel’fond, 1959, and Parry,1960)
In general, one knows that an invariant measure equivalent to theLebesgue measure exists for these transformations (Lasota and Yorke,1974).
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β-expansions and multiple tilings > Natural extensions and invariant measures
Invariant measures
The classic greedy β-transformation Tβ has the following properties.
The greedy β-transformation has an invariant measure that isequivalent to the Lebesgue measure on the unit interval [0, 1).(Renyi, 1957)
The density function is given by
hc : [0, 1)→ [0, 1) : x 7→ 1
F (β)
∞∑n=0
1
βn1[0,T n1)(x),
where F (β) is a normalizing constant. (Gel’fond, 1959, and Parry,1960)
In general, one knows that an invariant measure equivalent to theLebesgue measure exists for these transformations (Lasota and Yorke,1974).
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β-expansions and multiple tilings > Natural extensions and invariant measures
Natural extensions
A way to find an invariant measure is by studying the natural extensionof the dynamical system.
Consider the non-invertible system (X ,B,T ), where B is the Lebesgueσ-algebra on X . Then a version of the natural extension of (X ,B,T )is an invertible system (X , B, T ), such that
There is a map π : X → X that is surjective, measurable and suchthat π ◦ T = T ◦ π.
This system is the smallest in the sense of σ-algebras:∨n≥0 T n(π−1(B)) = B.
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β-expansions and multiple tilings > Natural extensions and invariant measures
Natural extensions
A way to find an invariant measure is by studying the natural extensionof the dynamical system.
Consider the non-invertible system (X ,B,T ), where B is the Lebesgueσ-algebra on X . Then a version of the natural extension of (X ,B,T )is an invertible system (X , B, T ), such that
There is a map π : X → X that is surjective, measurable and suchthat π ◦ T = T ◦ π.
This system is the smallest in the sense of σ-algebras:∨n≥0 T n(π−1(B)) = B.
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β-expansions and multiple tilings > Natural extensions and invariant measures
Pisot β’s
To be able to say more, we assume that the real number β > 1 hassome additional properties. Numbers with all these properties arecalled Pisot units.
β is an algebraic unit: it is a root of a minimal polynomial of theform xd − c1x
d−1 − · · · − cd , with ci ∈ Z for all i andcd ∈ {−1, 1}.Denote all the other roots of the polynomialxd − c1x
d−1 − . . .− cd by βj , then |βj | < 1 for all j .
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β-expansions and multiple tilings > Natural extensions and invariant measures
Pisot β’s
To be able to say more, we assume that the real number β > 1 hassome additional properties. Numbers with all these properties arecalled Pisot units.
β is an algebraic unit: it is a root of a minimal polynomial of theform xd − c1x
d−1 − · · · − cd , with ci ∈ Z for all i andcd ∈ {−1, 1}.Denote all the other roots of the polynomialxd − c1x
d−1 − . . .− cd by βj , then |βj | < 1 for all j .
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Contracting and expanding eigenspaces
Let β > 1 be a Pisot unit with minimal polynomialxd − c1x
d−1 − · · · − cd . Let β2, . . . , βd be the Galois conjugates of β.Consider the matrix M:
M =
c1 c2 · · · cd−1 cd
1 0 · · · 0 00 1 · · · 0 0...
.... . .
......
0 0 · · · 1 0
.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Contracting and expanding eigenspaces
Let β > 1 be a Pisot unit with minimal polynomialxd − c1x
d−1 − · · · − cd . Let β2, . . . , βd be the Galois conjugates of β.c1 c2 · · · cd−1 cd
1 0 · · · 0 00 1 · · · 0 0...
.... . .
......
0 0 · · · 1 0
βd−1j
βd−2j...1
=
c1β
d−1j + · · ·+ cd
βd−1j...βj
=
βd
j
βd−1j...βj
= βjvj .
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Contracting and expanding eigenspaces
Let β > 1 be a Pisot unit with minimal polynomialxd − c1x
d−1 − · · · − cd . Let β2, . . . , βd be the Galois conjugates of β.
M =
c1 c2 · · · cd−1 cd
1 0 · · · 0 00 1 · · · 0 0...
.... . .
......
0 0 · · · 1 0
.
Eigenvalues: β1 = β, β2, . . . , βd .Eigenvectors: v1, . . . , vd .| det M| = 1.
Let H be the hyperplane ofRd which is spanned by thereal and imaginary parts ofv2, . . . , vd .
Consider the space H + Rv1.M is expanding by a factorβ in the direction of v1 andcontracting by a factor 1/βon H.
(Rauzy (1982), Thurston (1989), Berthe and Siegel (2005))
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Contracting and expanding eigenspaces
Let β > 1 be a Pisot unit with minimal polynomialxd − c1x
d−1 − · · · − cd . Let β2, . . . , βd be the Galois conjugates of β.
M =
c1 c2 · · · cd−1 cd
1 0 · · · 0 00 1 · · · 0 0...
.... . .
......
0 0 · · · 1 0
.
Eigenvalues: β1 = β, β2, . . . , βd .Eigenvectors: v1, . . . , vd .| det M| = 1.
Let H be the hyperplane ofRd which is spanned by thereal and imaginary parts ofv2, . . . , vd .
Consider the space H + Rv1.M is expanding by a factorβ in the direction of v1 andcontracting by a factor 1/βon H.
(Rauzy (1982), Thurston (1989), Berthe and Siegel (2005))
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Contracting and expanding eigenspaces
Let β > 1 be a Pisot unit with minimal polynomialxd − c1x
d−1 − · · · − cd . Let β2, . . . , βd be the Galois conjugates of β.
M =
c1 c2 · · · cd−1 cd
1 0 · · · 0 00 1 · · · 0 0...
.... . .
......
0 0 · · · 1 0
.
Eigenvalues: β1 = β, β2, . . . , βd .Eigenvectors: v1, . . . , vd .| det M| = 1.
Let H be the hyperplane ofRd which is spanned by thereal and imaginary parts ofv2, . . . , vd .
Consider the space H + Rv1.M is expanding by a factorβ in the direction of v1 andcontracting by a factor 1/βon H.
(Rauzy (1982), Thurston (1989), Berthe and Siegel (2005))
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
Example: the golden mean
Take β > 1 such that β2 − β − 1 = 0. Then 1β = β − 1. So, β is an
algebraic unit. Then(− 1
β
)2−(− 1
β
)− 1 = (1− β)2 + (β − 1)− 1 = β2 − β − 1 = 0.
Hence, β2 = − 1β and β is a Pisot unit. We have
M =
(1 11 0
), v1 =
(β1
), v2 =
(− 1β
1
).
Then H = Rv2 and R2 is spanned by v1 and v2.
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β-expansions and multiple tilings > Natural extensions and invariant measures
Example: the tribonacci number
Take β > 1 such that β3 − β2 − β − 1 = 0. Since 1β = β2 − β − 1, β is
an algebraic unit. We have β2 ∈ C and β3 = β2. Also,
M =
1 1 11 0 00 1 0
, v1 =
β2
β1
, v2 =
β22
β2
1
, v3 =
β23
β3
1
.
Since v3 = v2, H is spanned by <(v2) and =(v2).
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β-expansions and multiple tilings > Natural extensions and invariant measures
Set-up
We now have the following set-up:
β > 1 is a Pisot unit.This gives a matrix M with eigenvalues β = β1, β2, . . . , βd andeigenvectors v1, . . . , vd .The space H is the hyperplane of Rd spanned by the real andimaginary parts of v2, . . . , vd . Every point in x ∈ Rd can bewritten as xv1 −
∑dj=2 yjvj with x ∈ R and yj ∈ C, 2 ≤ j ≤ d .
The transformation T : X → X is given by β, an allowable digitset A ⊂ Q(β) and a finite union of bounded intervals X . For eachdigit ai ∈ A, let Xi ⊆ X be the interval on which T is given byTx = βx − ai .For x ∈ X , b(x) denotes de digit sequence of x generated by T .The transformation T specifies a set of two-sided admissiblesequences
S = {· · · u−1u0u1u2 · · · | ∀n ∈ Z∃x ∈ X : unun+1 · · · = b(x)}.Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)T
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β-expansions and multiple tilings > Natural extensions and invariant measures
Set-up
We now have the following set-up:
β > 1 is a Pisot unit.This gives a matrix M with eigenvalues β = β1, β2, . . . , βd andeigenvectors v1, . . . , vd .The space H is the hyperplane of Rd spanned by the real andimaginary parts of v2, . . . , vd . Every point in x ∈ Rd can bewritten as xv1 −
∑dj=2 yjvj with x ∈ R and yj ∈ C, 2 ≤ j ≤ d .
The transformation T : X → X is given by β, an allowable digitset A ⊂ Q(β) and a finite union of bounded intervals X . For eachdigit ai ∈ A, let Xi ⊆ X be the interval on which T is given byTx = βx − ai .For x ∈ X , b(x) denotes de digit sequence of x generated by T .The transformation T specifies a set of two-sided admissiblesequences
S = {· · · u−1u0u1u2 · · · | ∀n ∈ Z∃x ∈ X : unun+1 · · · = b(x)}.Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)T
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β-expansions and multiple tilings > Natural extensions and invariant measures
Set-up
We now have the following set-up:
β > 1 is a Pisot unit.This gives a matrix M with eigenvalues β = β1, β2, . . . , βd andeigenvectors v1, . . . , vd .The space H is the hyperplane of Rd spanned by the real andimaginary parts of v2, . . . , vd . Every point in x ∈ Rd can bewritten as xv1 −
∑dj=2 yjvj with x ∈ R and yj ∈ C, 2 ≤ j ≤ d .
The transformation T : X → X is given by β, an allowable digitset A ⊂ Q(β) and a finite union of bounded intervals X . For eachdigit ai ∈ A, let Xi ⊆ X be the interval on which T is given byTx = βx − ai .For x ∈ X , b(x) denotes de digit sequence of x generated by T .The transformation T specifies a set of two-sided admissiblesequences
S = {· · · u−1u0u1u2 · · · | ∀n ∈ Z∃x ∈ X : unun+1 · · · = b(x)}.Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)T
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β-expansions and multiple tilings > Natural extensions and invariant measures
Set-up
We now have the following set-up:
β > 1 is a Pisot unit.This gives a matrix M with eigenvalues β = β1, β2, . . . , βd andeigenvectors v1, . . . , vd .The space H is the hyperplane of Rd spanned by the real andimaginary parts of v2, . . . , vd . Every point in x ∈ Rd can bewritten as xv1 −
∑dj=2 yjvj with x ∈ R and yj ∈ C, 2 ≤ j ≤ d .
The transformation T : X → X is given by β, an allowable digitset A ⊂ Q(β) and a finite union of bounded intervals X . For eachdigit ai ∈ A, let Xi ⊆ X be the interval on which T is given byTx = βx − ai .For x ∈ X , b(x) denotes de digit sequence of x generated by T .The transformation T specifies a set of two-sided admissiblesequences
S = {· · · u−1u0u1u2 · · · | ∀n ∈ Z∃x ∈ X : unun+1 · · · = b(x)}.Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)T
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β-expansions and multiple tilings > Natural extensions and invariant measures
Set-up
We now have the following set-up:
β > 1 is a Pisot unit.This gives a matrix M with eigenvalues β = β1, β2, . . . , βd andeigenvectors v1, . . . , vd .The space H is the hyperplane of Rd spanned by the real andimaginary parts of v2, . . . , vd . Every point in x ∈ Rd can bewritten as xv1 −
∑dj=2 yjvj with x ∈ R and yj ∈ C, 2 ≤ j ≤ d .
The transformation T : X → X is given by β, an allowable digitset A ⊂ Q(β) and a finite union of bounded intervals X . For eachdigit ai ∈ A, let Xi ⊆ X be the interval on which T is given byTx = βx − ai .For x ∈ X , b(x) denotes de digit sequence of x generated by T .The transformation T specifies a set of two-sided admissiblesequences
S = {· · · u−1u0u1u2 · · · | ∀n ∈ Z∃x ∈ X : unun+1 · · · = b(x)}.Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)T
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension space
We define the natural extension space by mapping the admissiblesequences into Rd .
For 1 ≤ j ≤ d , let Γj : Q(β)→ Q(βj) be given by Γj(β) = βj andΓj(q) = q for q ∈ Q.
Let w · u = · · ·w−1w0u1u2 · · · ∈ S. The map φ maps the sequence winto H:
φ(w) = φ(· · ·w−1w0) =d∑
j=2
∞∑n=0
w−nβnj vj .
Then ψ maps w · u into Rd :
ψ(w · u) = ψ(· · ·w−1w0u1u2 · · · ) =∞∑
n=1
un
βnv1 − φ(w).
Set X = ψ(S). This is the natural extension space.
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension space
We define the natural extension space by mapping the admissiblesequences into Rd .
For 1 ≤ j ≤ d , let Γj : Q(β)→ Q(βj) be given by Γj(β) = βj andΓj(q) = q for q ∈ Q.
Let w · u = · · ·w−1w0u1u2 · · · ∈ S. The map φ maps the sequence winto H:
φ(w) = φ(· · ·w−1w0) =d∑
j=2
∞∑n=0
w−nβnj vj .
Then ψ maps w · u into Rd :
ψ(w · u) = ψ(· · ·w−1w0u1u2 · · · ) =∞∑
n=1
un
βnv1 − φ(w).
Set X = ψ(S). This is the natural extension space.
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension space
We define the natural extension space by mapping the admissiblesequences into Rd .
For 1 ≤ j ≤ d , let Γj : Q(β)→ Q(βj) be given by Γj(β) = βj andΓj(q) = q for q ∈ Q.
Let w · u = · · ·w−1w0u1u2 · · · ∈ S. The map φ maps the sequence winto H:
φ(w) = φ(· · ·w−1w0) =d∑
j=2
∞∑n=0
w−nβnj vj .
Then ψ maps w · u into Rd :
ψ(w · u) = ψ(· · ·w−1w0u1u2 · · · ) =∞∑
n=1
un
βnv1 − φ(w).
Set X = ψ(S). This is the natural extension space.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension space
We define the natural extension space by mapping the admissiblesequences into Rd .
For 1 ≤ j ≤ d , let Γj : Q(β)→ Q(βj) be given by Γj(β) = βj andΓj(q) = q for q ∈ Q.
Let w · u = · · ·w−1w0u1u2 · · · ∈ S. The map φ maps the sequence winto H:
φ(w) = φ(· · ·w−1w0) =d∑
j=2
∞∑n=0
w−nβnj vj .
Then ψ maps w · u into Rd :
ψ(w · u) = ψ(· · ·w−1w0u1u2 · · · ) =∞∑
n=1
un
βnv1 − φ(w).
Set X = ψ(S). This is the natural extension space.
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension transformation
For the natural extension transformation T : X → X we want thefollowing.
T is a.e. invertible.
T preserves the dynamics of T .
T is invariant wrt the Lebesgue measure.
Partition X =⋃
i∈I Xi with Xi = {ψ(w · u) | u1 = ai}. For x ∈ X , write
x = xv1 −∑d
j=2 yjvj . If x ∈ Xi , take
Tx =
Tx︷ ︸︸ ︷(βx − ai ) v1 −
d∑j=2
(βjyj + Γj(ai ))vj
= Mx−d∑
j=1
Γj(ai )vj .
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension transformation
For the natural extension transformation T : X → X we want thefollowing.
T is a.e. invertible.
T preserves the dynamics of T .
T is invariant wrt the Lebesgue measure.
Partition X =⋃
i∈I Xi with Xi = {ψ(w · u) | u1 = ai}. For x ∈ X , write
x = xv1 −∑d
j=2 yjvj . If x ∈ Xi , take
Tx =
Tx︷ ︸︸ ︷(βx − ai ) v1 −
d∑j=2
(βjyj + Γj(ai ))vj
= Mx−d∑
j=1
Γj(ai )vj .
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension transformation
For the natural extension transformation T : X → X we want thefollowing.
T is a.e. invertible.
T preserves the dynamics of T .
T is invariant wrt the Lebesgue measure.
Partition X =⋃
i∈I Xi with Xi = {ψ(w · u) | u1 = ai}. For x ∈ X , write
x = xv1 −∑d
j=2 yjvj . If x ∈ Xi , take
Tx =
Tx︷ ︸︸ ︷(βx − ai ) v1 −
d∑j=2
(βjyj + Γj(ai ))vj
= Mx−d∑
j=1
Γj(ai )vj .
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension transformation
For the natural extension transformation T : X → X we want thefollowing.
T is a.e. invertible.
T preserves the dynamics of T .
T is invariant wrt the Lebesgue measure.
Partition X =⋃
i∈I Xi with Xi = {ψ(w · u) | u1 = ai}. For x ∈ X , write
x = xv1 −∑d
j=2 yjvj . If x ∈ Xi , take
Tx =
Tx︷ ︸︸ ︷(βx − ai ) v1 −
d∑j=2
(βjyj + Γj(ai ))vj
= Mx−d∑
j=1
Γj(ai )vj .
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
The natural extension transformation
For the natural extension transformation T : X → X we want thefollowing.
T is a.e. invertible.
T preserves the dynamics of T .
T is invariant wrt the Lebesgue measure.
Partition X =⋃
i∈I Xi with Xi = {ψ(w · u) | u1 = ai}. For x ∈ X , write
x = xv1 −∑d
j=2 yjvj . If x ∈ Xi , take
Tx =
Tx︷ ︸︸ ︷(βx − ai ) v1 −
d∑j=2
(βjyj + Γj(ai ))vj
= Mx−d∑
j=1
Γj(ai )vj .
Charlene Kalle, joint work with Wolfgang Steiner (PhD supervisor Karma Dajani)TH
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β-expansions and multiple tilings > Natural extensions and invariant measures
An invariant measure for T
The Lebesgue measure on λd is invariant for T . Let π : X → X begiven by π
(xv1 −
∑dj=2 yjvj
)= x .
Define the measure µ on (X ,L) by µ(E ) = (λd ◦ π−1)(E ) for eachE ∈ L. Then µ is invariant for T .
Note: Since T is only invertible a.e. we need to remove the right setsof measure zero everywhere.
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β-expansions and multiple tilings > Natural extensions and invariant measures
An example: the golden mean
Take β to be the golden mean and T the classic greedyβ-transformation, then A = {0, 1}. Recall that
M =
(1 11 0
), v1 =
(β1
), v2 =
(− 1β
1
).
0 1β
1
10
X0 X1
v1
v2
T X0
T X1v1
v2
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β-expansions and multiple tilings > Natural extensions and invariant measures
An example: the smallest Pisot number
Let β be the real solution of x3 − x − 1 = 0. This is the smallest Pisot
number. Take A = {−1, 0, 1}, X−1 =[− β7
β8−1,− β6
β8−1
),
X0 =[− β6
β8−1, β6
β8−1
)and X1 =
[β6
β8−1, β7
β8−1
). Then T is a minimal
weight transformation.
−1 0 1
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β-expansions and multiple tilings > Natural extensions and invariant measures
An example: the tribonacci number
Let β be the tribonacci number. Take A = {−1, 0, 1},X−1 =
[− β
β+1 ,−1
β+1
), X0 =
[− 1
β+1 ,1
β+1
)and X1 =
[1
β+1 ,ββ+1
).
Then T is a minimal weight transformation.
−1 0 1
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β-expansions and multiple tilings > Multiple tilings
Multiple tilings
Under a certain condition we can say more about the invariantmeasure for T given by its natural extension. Moreover, this conditionallows us to construct a tiling of the space H. A tiling is the following.
Start with a finite set of prototiles in H, compact sets that are theclosure of their interior. D = {D1, . . . ,Dk}.
A tile is a translation of a prototile. Tx = Di + vx for some vectorvx ∈ H.
Let m ≥ 1. A multiple tiling of degree m of H is a covering of H suchthat almost all elements are in exactly m different tiles.
A tiling of H is a multiple tiling of degree 1.
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β-expansions and multiple tilings > Multiple tilings
Tilings
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β-expansions and multiple tilings > Multiple tilings
A finite set of prototiles
For each x ∈ X , let Dx = {φ(w) |w · b(x) ∈ S}. Then each set Dx iscompact and
X =⋃x∈X
xv1 +Dx .
We will construct a multiple tiling of H with {Dx | x ∈ Z[β] ∩ X} asthe set of prototiles. Therefore, we would like to have only finitelymany different sets Dx .
0 1β
1
10
X0 X1
v1
v2
T X0
T X1v1
v2
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β-expansions and multiple tilings > Multiple tilings
A finite set of prototiles
Recall the transformation T :
γ0 γ1
Tγ2 γ3
a2a1a0
γ0 γ1
T
γ2 γ3
a2a1a0
For γi , let ni be the minimal k such that T kγi = T kγi with ni =∞ ifthis doesn’t happen.
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β-expansions and multiple tilings > Multiple tilings
A finite set of prototiles
Suppose that A = {a0, . . . , am}.
Theorem
If the set
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
is finite, then there are only finitely many different sets Dx , x ∈ X .
Under this condition the density of the invariant measure for T ,µ = λd ◦ π−1 is a finite sum of indicator functions. Let D1, . . . ,Dκ bethese sets and X ′1, . . . ,X
′κ the corresponding subsets of X . Then
µ([s, t)) = c
∫[s,t)
κ∑k=1
λd−1(Dk)1X ′kdλ.
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β-expansions and multiple tilings > Multiple tilings
An example: the classic greedy tribonacci-transformation
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
0 1/β 1
γ2γ1γ0 ⋃2i=0{γi} = {0, 1
β , 1}.{γi ∈ X | i 6= 0} = {1/β}.T k(
1β
)= 0 for all k ≥ 1.
T(
1β
)= 1, T 2
(1β
)= β− 1, T 3
(1β
)= 1
β .
So, n1 =∞, but γ1 is periodic for T .
V = {0, 1β , β − 1, 1} is a finite set.
This gives 3 different prototiles Dx .
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β-expansions and multiple tilings > Multiple tilings
An example: the classic greedy tribonacci-transformation
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
0 1/β 1
γ2γ1γ0 ⋃2i=0{γi} = {0, 1
β , 1}.{γi ∈ X | i 6= 0} = {1/β}.T k(
1β
)= 0 for all k ≥ 1.
T(
1β
)= 1, T 2
(1β
)= β− 1, T 3
(1β
)= 1
β .
So, n1 =∞, but γ1 is periodic for T .
V = {0, 1β , β − 1, 1} is a finite set.
This gives 3 different prototiles Dx .
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β-expansions and multiple tilings > Multiple tilings
An example: the classic greedy tribonacci-transformation
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
0 1/β 1
γ2γ1γ0 ⋃2i=0{γi} = {0, 1
β , 1}.{γi ∈ X | i 6= 0} = {1/β}.T k(
1β
)= 0 for all k ≥ 1.
T(
1β
)= 1, T 2
(1β
)= β− 1, T 3
(1β
)= 1
β .
So, n1 =∞, but γ1 is periodic for T .
V = {0, 1β , β − 1, 1} is a finite set.
This gives 3 different prototiles Dx .
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β-expansions and multiple tilings > Multiple tilings
An example: the classic greedy tribonacci-transformation
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
0 1/β 1
γ2γ1γ0 ⋃2i=0{γi} = {0, 1
β , 1}.{γi ∈ X | i 6= 0} = {1/β}.T k(
1β
)= 0 for all k ≥ 1.
T(
1β
)= 1, T 2
(1β
)= β− 1, T 3
(1β
)= 1
β .
So, n1 =∞, but γ1 is periodic for T .
V = {0, 1β , β − 1, 1} is a finite set.
This gives 3 different prototiles Dx .
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β-expansions and multiple tilings > Multiple tilings
An example: the classic greedy tribonacci-transformation
V =m+1⋃i=0
{γi} ∪⋃
1≤k<ni ,γi∈X ,i 6=0
{T kγi , Tkγi}
0 1/β 1
γ2γ1γ0 ⋃2i=0{γi} = {0, 1
β , 1}.{γi ∈ X | i 6= 0} = {1/β}.T k(
1β
)= 0 for all k ≥ 1.
T(
1β
)= 1, T 2
(1β
)= β− 1, T 3
(1β
)= 1
β .
So, n1 =∞, but γ1 is periodic for T .
V = {0, 1β , β − 1, 1} is a finite set.
This gives 3 different prototiles Dx .
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β-expansions and multiple tilings > Multiple tilings
The translation vectors
Suppose that V is a finite set. The set {Dx | x ∈ X} is finite and is theset of prototiles. We now want a set of translation vectors, so that
all the translates of Dx together cover the whole space H, and
there is an m ≥ 1 such that a.e. point in H is in exactly mtranslates of prototiles.
So the set of translation vectors must be big enough, but not too big.
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β-expansions and multiple tilings > Multiple tilings
The translation vectors
Suppose that V is a finite set. The set {Dx | x ∈ X} is finite and is theset of prototiles. We now want a set of translation vectors, so that
all the translates of Dx together cover the whole space H, and
there is an m ≥ 1 such that a.e. point in H is in exactly mtranslates of prototiles.
So the set of translation vectors must be big enough, but not too big.
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β-expansions and multiple tilings > Multiple tilings
The translation vectors
Suppose that V is a finite set. The set {Dx | x ∈ X} is finite and is theset of prototiles. We now want a set of translation vectors, so that
all the translates of Dx together cover the whole space H, and
there is an m ≥ 1 such that a.e. point in H is in exactly mtranslates of prototiles.
So the set of translation vectors must be big enough, but not too big.
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β-expansions and multiple tilings > Multiple tilings
The multiple tiling
Define the function Φ : Q(β)→ H by Φ(x) =∑d
j=2 Γj(x)vj .
For x ∈ Z[β] ∩ X , define the tiles Tx = Φ(x) +Dx .
Theorem
There is an m ≥ 1, such that the set {Tx}x∈Z[β]∩X is a multiple tilingof degree m of H.
Pisot conjecture (Akiyama, 2002 and Sidorov, 2003)
If β is a Pisot number and T is the classic greedy β-transformation,then this construction gives a tiling of the space H.
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β-expansions and multiple tilings > Multiple tilings
An example: the Rauzy tiling (Rauzy, 1982)
Let β be the tribonacci number and T the classic greedyβ-transformation.
0 1β
1
101
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β-expansions and multiple tilings > Multiple tilings
An example: the Rauzy tiling (Rauzy, 1982)
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β-expansions and multiple tilings > Multiple tilings
A tiling: the tribonacci number
Let β be the tribonacci number. Take A = {−1, 0, 1},X−1 =
[− β
β+1 ,−1
β+1
), X0 =
[− 1
β+1 ,1
β+1
)and X1 =
[1
β+1 ,ββ+1
).
Then T is a minimal weight transformation.
−1 0 1
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β-expansions and multiple tilings > Multiple tilings
A tiling: the tribonacci number
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β-expansions and multiple tilings > Multiple tilings
A tiling: the smallest Pisot number
Let β be the real solution of x3 − x − 1 = 0. This is the smallest Pisot
number. Take A = {−1, 0, 1}, X−1 =[− β7
β8−1,− β6
β8−1
),
X0 =[− β6
β8−1, β6
β8−1
)and X1 =
[β6
β8−1, β7
β8−1
). Then T is a minimal
weight transformation.
−1 0 1
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β-expansions and multiple tilings > Multiple tilings
A tiling: the smallest Pisot number
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
Let β be the tribonacci number. Take A = {−1, 0, 1},X−1 =
[− 1
2 ,−12β
), X0 =
[− 1
2β ,12β
)and X1 =
[12β ,
12
).
−1 0 1
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
T .(101)�
T .(110)�
T .0(101)�
T .(011)�
T .1(110)�
T .10(101)�
T .00(101)� T .0(011)
�
T .11(110)�
T .01(110)�
T .110(101)�
T .10(011)�
T .101(110)�
T .001(110)�
T .1110(101)�
T .110(011)�
T .(101)�
T .(110)�T .0(101)�
T .(011)�T .1(110)�
T .10(101)�T .00(101)�T .0(011)
�
T .11(110)�T .01(110)�
T .110(101)�
T .10(011)�
T .101(110)�T .001(110)�
T .1110(101)�
T .110(011)�
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
Consider 2 tiles T1− 1β
and T 1β3
.
Take a point y from the yellowball in T1− 1
β.
Then y = Φ(1 − 1β ) + φ(w) for
some w such that w ·b(1− 1β ) ∈
S.Show that there is a sequencew ′, such that w ′ ·b( 1
β3 ) and y =
Φ( 1β3 ) + φ(w ′).
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
Consider 2 tiles T1− 1β
and T 1β3
.
Take a point y from the yellowball in T1− 1
β.
Then y = Φ(1 − 1β ) + φ(w) for
some w such that w ·b(1− 1β ) ∈
S.Show that there is a sequencew ′, such that w ′ ·b( 1
β3 ) and y =
Φ( 1β3 ) + φ(w ′).
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
Consider 2 tiles T1− 1β
and T 1β3
.
Take a point y from the yellowball in T1− 1
β.
Then y = Φ(1 − 1β ) + φ(w) for
some w such that w ·b(1− 1β ) ∈
S.Show that there is a sequencew ′, such that w ′ ·b( 1
β3 ) and y =
Φ( 1β3 ) + φ(w ′).
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the tribonacci number
Consider 2 tiles T1− 1β
and T 1β3
.
Take a point y from the yellowball in T1− 1
β.
Then y = Φ(1 − 1β ) + φ(w) for
some w such that w ·b(1− 1β ) ∈
S.Show that there is a sequencew ′, such that w ′ ·b( 1
β3 ) and y =
Φ( 1β3 ) + φ(w ′).
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the smallest Pisot number
Let β be the smallest Pisot number. Take A = {−1, 0, 1},X−1 =
[− 1
2 ,−12β
), X0 =
[− 1
2β ,12β
)and X1 =
[12β ,
12
).
−1 0 1
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β-expansions and multiple tilings > Multiple tilings
A double tiling: the smallest Pisot number
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β-expansions and multiple tilings > Multiple tilings
Properties of the multiple tiling
The tiles that contain the origin are precisely the tiles Tx forwhich x has a purely periodic digit sequence b(x).
The multiple tiling is self-replicating.
The multiple tiling is quasi-periodic: ∀r > 0 ∃R such that for ally, y′ ∈ H, the local configuration that we see in the ball B(y, r)also occurs in the ball B(y′,R).
If the multiple tiling is a tiling, then the closure of the naturalextension domain gives a tiling of the torus.
If we have a tiling and every set Dx contains the origin, then thereis a direct way to determine the n-th digit of the expansions.
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β-expansions and multiple tilings > Multiple tilings
Properties of the multiple tiling
The tiles that contain the origin are precisely the tiles Tx forwhich x has a purely periodic digit sequence b(x).
The multiple tiling is self-replicating.
The multiple tiling is quasi-periodic: ∀r > 0 ∃R such that for ally, y′ ∈ H, the local configuration that we see in the ball B(y, r)also occurs in the ball B(y′,R).
If the multiple tiling is a tiling, then the closure of the naturalextension domain gives a tiling of the torus.
If we have a tiling and every set Dx contains the origin, then thereis a direct way to determine the n-th digit of the expansions.
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β-expansions and multiple tilings > Multiple tilings
Properties of the multiple tiling
The tiles that contain the origin are precisely the tiles Tx forwhich x has a purely periodic digit sequence b(x).
The multiple tiling is self-replicating.
The multiple tiling is quasi-periodic: ∀r > 0 ∃R such that for ally, y′ ∈ H, the local configuration that we see in the ball B(y, r)also occurs in the ball B(y′,R).
If the multiple tiling is a tiling, then the closure of the naturalextension domain gives a tiling of the torus.
If we have a tiling and every set Dx contains the origin, then thereis a direct way to determine the n-th digit of the expansions.
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β-expansions and multiple tilings > Multiple tilings
Properties of the multiple tiling
The tiles that contain the origin are precisely the tiles Tx forwhich x has a purely periodic digit sequence b(x).
The multiple tiling is self-replicating.
The multiple tiling is quasi-periodic: ∀r > 0 ∃R such that for ally, y′ ∈ H, the local configuration that we see in the ball B(y, r)also occurs in the ball B(y′,R).
If the multiple tiling is a tiling, then the closure of the naturalextension domain gives a tiling of the torus.
If we have a tiling and every set Dx contains the origin, then thereis a direct way to determine the n-th digit of the expansions.
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β-expansions and multiple tilings > Multiple tilings
Properties of the multiple tiling
The tiles that contain the origin are precisely the tiles Tx forwhich x has a purely periodic digit sequence b(x).
The multiple tiling is self-replicating.
The multiple tiling is quasi-periodic: ∀r > 0 ∃R such that for ally, y′ ∈ H, the local configuration that we see in the ball B(y, r)also occurs in the ball B(y′,R).
If the multiple tiling is a tiling, then the closure of the naturalextension domain gives a tiling of the torus.
If we have a tiling and every set Dx contains the origin, then thereis a direct way to determine the n-th digit of the expansions.
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