Download (−β)-expansion of real numbers

(−β)-expansion of real numbers
Shunji Ito & Taizo Sadahiro
Review of β-expansions
Let β > 1 be a real number. A β -representation of a real
number x is an expression of the form,
k
x = x−k β + x−k+1 β
k−1
x1 x2
+ 2 + ··· ,
+ · · · + x0 +
β
β
where k ≥ 0 is a certain integer and xi > 0 for i ≥ −k . It is
denoted by
x = (x−k x−k+1 · · · x0 . x1 x2 · · · )β .
(−β)-expansion of real numbers – p.1
Review of β-expansions
The β -transformation Tβ : [0, 1) → [0, 1) is defined by
Tβ (x) = {βx} = βx
mod 1.
β = 2.3
(−β)-expansion of real numbers – p.2
Review of β-expansions
Then, for each x ∈ [0, 1), we have a particular
β -representation
x = (0 . x1 x2 · · · )β .
where xi = ⌊βTβi−1 (x)⌋ for i ≥ 1.
We call this representation the β -expansion of x.
(−β)-expansion of real numbers – p.3
Review of β-expansions
A sequence (x1 , x2 , . . .) is admissible if there exists
x ∈ [0, 1) such that
x = (0 . x1 x2 . . .)β
is the β -expansion of x.
▽(−β)-expansion of real numbers – p.4
Review of β-expansions
A sequence (x1 , x2 , . . .) is admissible if there exists
x ∈ [0, 1) such that
x = (0 . x1 x2 . . .)β
is the β -expansion of x.
Theorem 2 (Parry). A sequence (x1 , x2 , . . .) is admissible if and
only if
(x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
where the sequence d∗ (1, β) is defined as follows.
(−β)-expansion of real numbers – p.4
Review of β-expansions
β -expansion of the fractional part {β} of β :
{β} = β − ⌊β⌋ = (0 . d1 d2 . . .).
Then we have a β -representation of 1:
1 = (0 . ⌊β⌋d1 d2 · · · )β .
∗
d (1, β) :=
(
(⌊β⌋, d1 , d2 , . . . , di−1 , di − 1)
(⌊β⌋, d1 , d2 , . . .)
0 = di+1 = di+2 = · · ·
otherwise
(−β)-expansion of real numbers – p.5
Review of β-expansions
Theorem 3 (Renyi). The β -transformation is ergodic with unique
invariant measure equivalent to the Lebesque measure.
Theorem 4 (Parry). Let hβ : [0, 1) → R be defined by
X 1
,
hβ (x) =
n
β
x≤sn
where s0 = 1 and sn = Tβn−1 ({β}) for n ≥ 1. Then the measure
dµ = hβ dx is invariant under Tβ where dx denotes the Lebesgue
measure.
(−β)-expansion of real numbers – p.6
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
▽(−β)-expansion of real numbers – p.7
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
The value of β -transformation can be expressed as,
Tβ (x) = {βx}
▽(−β)-expansion of real numbers – p.7
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
The value of β -transformation can be expressed as,
Tβ (x) = {βx}
= {βx−0}
▽(−β)-expansion of real numbers – p.7
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
The value of β -transformation can be expressed as,
Tβ (x) = {βx}
= {βx−0}+0
▽(−β)-expansion of real numbers – p.7
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
The value of β -transformation can be expressed as,
Tβ (x) = {βx}
= {βx−0}+0
0 is the left endpoint of [0, 1).
▽(−β)-expansion of real numbers – p.7
Trivial remarks
Parry’s criteria for the admissibility can be writen as,
(0, 0, 0, · · · ) lex (x1 , x2 , . . .) ≺lex d∗ (1, β), ∀i ≥ 1.
The value of β -transformation can be expressed as,
Tβ (x) = {βx}
= {βx−0}+0
0 is the left endpoint of [0, 1).
xi = ⌊βT i−1 (x)−0⌋
(−β)-expansion of real numbers – p.7
Definition: (−β)-representation
β > 1 A (−β)-representation of a real number x is an
expression of the form,
k
x = x−k (−β) +x−k+1 (−β)
k−1
x2
x1
+· · · ,
+
+· · ·+x0 +
2
(−β) (−β)
where k ≥ 0 is a certain integer and xi > 0 for i ≥ −k .
▽(−β)-expansion of real numbers – p.8
Definition: (−β)-representation
β > 1 A (−β)-representation of a real number x is an
expression of the form,
k
x = x−k (−β) +x−k+1 (−β)
k−1
x2
x1
+· · · ,
+
+· · ·+x0 +
2
(−β) (−β)
where k ≥ 0 is a certain integer and xi > 0 for i ≥ −k .
It is denoted by
x = (x−k x−k+1 · · · x0 . x1 x2 · · · )−β .
(−β)-expansion of real numbers – p.8
Definition: (−β)-transformation
h
β
1
Iβ = [lβ , rβ ) = − β+1
, β+1
.
▽(−β)-expansion of real numbers – p.9
Definition: (−β)-transformation
h
β
1
Iβ = [lβ , rβ ) = − β+1
, β+1
.
The (−β)-transformation T−β on Iβ is defined by
T−β (x) = {−βx−lβ }+lβ
▽(−β)-expansion of real numbers – p.9
Definition: (−β)-transformation
h
β
1
Iβ = [lβ , rβ ) = − β+1
, β+1
.
The (−β)-transformation T−β on Iβ is defined by
T−β (x) = {−βx−lβ }+lβ
= −βx − −βx +
β
β+1
(−β)-expansion of real numbers – p.9
Definition
β = 2.3
1
β+1
β
− β+1
1
β+1
β
− β+1
▽(−β)-expansion of real numbers – p.10
Definition
β = 2.3
1
β+1
β
− β+1
1
β+1
β
− β+1
▽(−β)-expansion of real numbers – p.10
Definition
β = 2.3
1
β+1
β2
− β+1
1
0
1
β+1
β
− β+1
(−β)-expansion of real numbers – p.10
Definition
Then, for each x ∈ Iβ , we have a particular
(−β)-representation
x = ( . x1 x2 · · · )−β .
i−1
where xi = ⌊−βT−β
(x)−lβ ⌋ for i ≥ 1. We call this
representation the (−β)-expansion of x.
(−β)-expansion of real numbers – p.11
Definition
For a real number x not contained in Iβ , there is an
integer d such that x/(−β)d ∈ Iβ , hence we have the
(−β)-expansion of x:
x = (x−d+1 x−d+2 · · · x0 . x1 x2 · · · )−β
i−1
x
where x−d+i = ⌊−βT−β
( (−β)
d) +
(1)
β
β+1 ⌋.
(−β)-expansion of real numbers – p.12
Examples
Example 1. β = 2
▽(−β)-expansion of real numbers – p.13
Examples
Example 2. β = 2
2 = (110.)−2 ,
3 = (111.)−2 ,
4 = (100.)−2 ,
..
.
100 = (110100100.)−2 ,
..
.
▽(−β)-expansion of real numbers – p.13
Examples
Example 3. β = 2
2 = (110.)−2 ,
3 = (111.)−2 ,
4 = (100.)−2 ,
..
.
100 = (110100100.)−2 ,
..
.
−1 = (11 . )−2 ,
−2 = (10.)−2 ,
−3 = (1101.)−2 ,
..
.
−100 = (11101100.)−2
..
.
▽(−β)-expansion of real numbers – p.13
Examples
Example 4. β = 2
2 = (110.)−2 ,
3 = (111.)−2 ,
4 = (100.)−2 ,
..
.
100 = (110100100.)−2 ,
..
.
−1 = (11 . )−2 ,
−2 = (10.)−2 ,
−3 = (1101.)−2 ,
..
.
−100 = (11101100.)−2
..
.
2/3 = (1.111111 · · · )−2 , 1/5 = (.011101110111 · · · )−2 .
▽(−β)-expansion of real numbers – p.13
Examples
Example 5. β = 2
2 = (110.)−2 ,
3 = (111.)−2 ,
4 = (100.)−2 ,
−1 = (11 . )−2 ,
−2 = (10.)−2 ,
−3 = (1101.)−2 ,
..
.
..
.
100 = (110100100.)−2 ,
..
.
−100 = (11101100.)−2
..
.
2/3 = (1.111111 · · · )−2 , 1/5 = (.011101110111 · · · )−2 .
−2/3 = (0.22222 · · · )−2
▽(−β)-expansion of real numbers – p.13
Examples
Example 6. β = 2
2 = (110.)−2 ,
3 = (111.)−2 ,
4 = (100.)−2 ,
..
.
..
.
100 = (110100100.)−2 ,
..
.
−1 = (11 . )−2 ,
−2 = (10.)−2 ,
−3 = (1101.)−2 ,
−100 = (11101100.)−2
..
.
2/3 = (1.111111 · · · )−2 , 1/5 = (.011101110111 · · · )−2 .
−2/3 = (0.22222 · · · )−2 = (0.10101010 · · · )−2
.
(−β)
-expansion of real numbers – p.13
Examples
Example 7. β > 0 satisfies β 3 − β 2 − β − 1 = 0.
2 = (111 . 1)−β ,
3 = (100 . 111001)−β ,
4 = (101 . 111001)−β ,
..
.
−1 = (11 . 001)−β ,
−2 = (10 . 001)−β ,
..
.
−100 = (1100010010 . 0100010000010001
100 = (111000110 . 00001100101111)
. −β ,
..
.
..
(−β)-expansion of real numbers – p.14
Admissible sequences
We say an integer sequence (x1 , x2 , . . .) is
(−β)-admissible, if there exists a real number x ∈ Iβ such
that x = ( . x1 x2 · · · )−β is a (−β)-expansion.
▽(−β)-expansion of real numbers – p.15
Admissible sequences
We say an integer sequence (x1 , x2 , . . .) is
(−β)-admissible, if there exists a real number x ∈ Iβ such
that x = ( . x1 x2 · · · )−β is a (−β)-expansion.
We define an order ≺ on the sequences of integers. Let
(x1 , x2 , . . .) and (y1 , y2 , . . .) be two integer sequences .
Then
(x1 , x2 , . . .) ≺ (y1 , y2 , . . .)
if there exsists an integer k ≥ 1 such that xi = yi for i < k
and
(−1)k (xk − yk ) < 0.
(−β)-expansion of real numbers – p.15
Admissible sequences
Let
−β
lβ =
= ( . b∗1 b∗2 · · · )−β
β+1
be the (−β)-expansion of the left endpoint of Iβ . We call
(b∗1 , b∗2 , . . .) the lower sequence of −β .
▽(−β)-expansion of real numbers – p.16
Admissible sequences
Let
−β
lβ =
= ( . b∗1 b∗2 · · · )−β
β+1
be the (−β)-expansion of the left endpoint of Iβ . We call
(b∗1 , b∗2 , . . .) the lower sequence of −β .
Then, we have a (−β)-representation of the right
1
:
endpoint rβ = β+1
1
= ( . 0b∗1 b∗2 b∗3 · · · )−β ,
rβ =
β+1
which is not the (−β)-expansion.
(−β)-expansion of real numbers – p.16
Admissible sequences
Proposition 1. If an integer sequence (x1 , x2 , . . .) is
(−β)-admissible, then
(b∗1 , b∗2 , . . .) (x∗n+1 , x∗n+2 , . . .) ≺ (0, b∗1 , b∗2 , . . .), ∀n ≥ 0.
▽(−β)-expansion of real numbers – p.17
Admissible sequences
Proposition 2. If an integer sequence (x1 , x2 , . . .) is
(−β)-admissible, then
(b∗1 , b∗2 , . . .) (x∗n+1 , x∗n+2 , . . .) ≺ (0, b∗1 , b∗2 , . . .), ∀n ≥ 0.
The converse of Proposition is not generally true.
▽(−β)-expansion of real numbers – p.17
Admissible sequences
Proposition 3. If an integer sequence (x1 , x2 , . . .) is
(−β)-admissible, then
(b∗1 , b∗2 , . . .) (x∗n+1 , x∗n+2 , . . .) ≺ (0, b∗1 , b∗2 , . . .), ∀n ≥ 0.
The converse of Proposition is not generally true.
Example 10. β = 2, (b∗1 , b∗2 , b∗3 , . . .) = (2, 2, 2, . . .)
(2, 2, 2, . . .) (0, 1, 0, 1, 0, 1, 0, 1, . . .) ≺ (0, 2, 2, 2 . . .)
But,
(0.010101 · · · )−2
1
= 6∈ I−β .
3
(−β)-expansion of real numbers – p.17
Admissible sequences
The upper sequence (c∗1 , c∗2 , . . .) of −β :
▽(−β)-expansion of real numbers – p.18
Admissible sequences
The upper sequence (c∗1 , c∗2 , . . .) of −β :
If the lower sequence (b∗1 , b∗2 , . . .) is purely periodic with an
odd period q , i.e., b∗i+q = b∗i for all i ≥ 1, then we define


0
c∗i = b∗i−1 mod (q+1)

 ∗
bq+1 − 1
i ≡ 1 mod (q + 1)
i 6≡ 0, 1 mod (q + 1),
i ≡ 0 mod (q + 1).
That is, (c∗1 , c∗2 , . . .) = (c∗1 , c∗2 , . . . c∗q+1 ) = (0, b∗1 , b∗2 , . . . , b∗q − 1)
Otherwise, we define (c∗1 , c∗2 , . . .) = (0, b∗1 , b∗2 , . . .).
(−β)-expansion of real numbers – p.18
Admissible sequences
Example 11. Let β be the real root of X 3 − 2X 2 + X − 1 = 0.
Then,
(b∗1 , b∗2 , . . .) = (1, 0, 1).
Therefore −β has the upper sequence
(c∗1 , c∗2 , . . .) = (0, 1, 0, 0).
(−β)-expansion of real numbers – p.19
Admissible Sequences
Theorem 5. A sequence (x1 , x2 , . . .) of non-negative integers is
(−β)-admissible if and only if
(b∗1 , b∗2 , . . .) (xn+1 , xn+2 , . . .) ≺ (c∗1 , c∗2 , . . .) for all n ≥ 0.
(−β)-expansion of real numbers – p.20
Invariant measure
Theorem 6. Let h−β : Iβ → R be defined by
h−β (x) =
X
x≥sn
1
,
n
(−β)
where s0 = lβ , and si = T−β (si−1 ). Then the measure
dµ = h−β dλ is invariant under T−β , where dλ denotes the
Lebesgue measure.
(−β)-expansion of real numbers – p.21
Example
Let β be the minimal Pisot number. (β 3 = β + 1)
▽(−β)-expansion of real numbers – p.22
Example
Let β be the minimal Pisot number. (β 3 = β + 1)
lβ = s0 < s3 < s1 < s2 ,
s3 = s4 = · · · .
▽(−β)-expansion of real numbers – p.22
Example
Let β be the minimal Pisot number. (β 3 = β + 1)
lβ = s0 < s3 < s1 < s2 ,
s3 = s4 = · · · .
s0 ∼ s3 ∼ s1 ∼ s2 ∼
√
√
√
√
1
√
√
1
−β
√
1
β2
√
√
√
1
− β3
√
√
√
1
β4
..
.
h−β
..
.
1
..
.
1
β
..
.
0
..
.
1
β2
▽(−β)-expansion of real numbers – p.22
Example
Let β be the minimal Pisot number. (β 3 = β + 1)
lβ = s0 < s3 < s1 < s2 ,
s3 = s4 = · · · .
s0 ∼ s3 ∼ s1 ∼ s2 ∼
√
√
√
√
1
√
√
1
−β
√
1
β2
√
√
√
µ is not equivalent to
1
− β3
the Lebesgue mea√
√
√
1
sure.
β4
..
..
..
..
..
.
.
.
.
.
1
1
h−β
1
0
β
β2
▽(−β)-expansion of real numbers – p.22
Example
Let β be the minimal Pisot number. (β 3 = β + 1)
lβ = s0 < s3 < s1 < s2 ,
s3 = s4 = · · · .
s0 ∼ s3 ∼ s1 ∼ s2 ∼
√
√
√
√
1
√
√
1
−β
√
1
β2
√
√
√
µ is not equivalent to
1
− β3
the Lebesgue mea√
√
√
1
sure.
β4
..
..
..
..
..
.
.
.
.
.
1
1
h−β
1
0
β
β2
(−β)-expansion of real numbers – p.22
Thank you very much.
(−β)-expansion of real numbers – p.23