( ) -expansion of real numbers Shunji Ito & Taizo Sadahiro - - PowerPoint PPT Presentation

(−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,

x = x−kβk + x−k+1βk−1 + · · · + x0 + x1 β + x2 β2 + · · · ,

where k ≥ 0 is a certain integer and xi > 0 for i ≥ −k. It is denoted by

x = (x−kx−k+1 · · · x0 . x1x2 · · · )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 . x1x2 · · · )β.

where xi = ⌊βT i−1

β

(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 . x1x2 . . .)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 . x1x2 . . .)β

is the β-expansion of x.

Theorem 2 (Parry). A sequence (x1, x2, . . .) is admissible if and

• nly 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 . d1d2 . . .).

Then we have a β-representation of 1:

1 = (0 . ⌊β⌋d1d2 · · · )β. d∗(1, β) :=

• (⌊β⌋, d1, d2, . . . , di−1, di − 1)

0 = di+1 = di+2 = · · · (⌊β⌋, d1, d2, . . .)

• therwise

(−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

hβ(x) =

• x≤sn

1 βn,

where s0 = 1 and sn = T n−1

β

({β}) for n ≥ 1. Then the measure dµ = hβdx 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,

x = x−k(−β)k+x−k+1(−β)k−1+· · ·+x0+ x1 (−β)+ x2 (−β)2 +· · · ,

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,

x = x−k(−β)k+x−k+1(−β)k−1+· · ·+x0+ x1 (−β)+ x2 (−β)2 +· · · ,

where k ≥ 0 is a certain integer and xi > 0 for i ≥ −k. It is denoted by

x = (x−kx−k+1 · · · x0 . x1x2 · · · )−β.

(−β)-expansion of real numbers – p.8

Definition: (−β)-transformation

Iβ = [lβ, rβ) =

• − β

β+1, 1 β+1

• .

▽(−β)-expansion of real numbers – p.9

Definition: (−β)-transformation

Iβ = [lβ, rβ) =

• − β

β+1, 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

Iβ = [lβ, rβ) =

• − β

β+1, 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

1 β+1 1 β+1

−

β β+1

−

β β+1

β = 2.3

▽(−β)-expansion of real numbers – p.10

Definition

1 β+1 1 β+1

−

β β+1

−

β β+1

β = 2.3

▽(−β)-expansion of real numbers – p.10

Definition

1 β+1 1 β+1

−

β β+1

−

β β+1

β = 2.3

1 2

(−β)-expansion of real numbers – p.10

Definition

Then, for each x ∈ Iβ, we have a particular

(−β)-representation x = ( . x1x2 · · · )−β.

where xi = ⌊−βT i−1

−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+1x−d+2 · · · x0 . x1x2 · · · )−β

(1)

where x−d+i = ⌊−βT i−1

−β ( x (−β)d) + 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,

. . .

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

▽(−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 β3 − β2 − β − 1 = 0.

2 = (111 . 1)−β, 3 = (100 . 111001)−β, 4 = (101 . 111001)−β,

. . .

100 = (111000110.00001100101111)−β,

. . .

−1 = (11 . 001)−β, −2 = (10 . 001)−β,

. . .

−100 = (1100010010 . 01000100000100

. . .

(−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 = ( . x1x2 · · · )−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 = ( . x1x2 · · · )−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

lβ = −β β + 1 = ( . b∗

1b∗ 2 · · · )−β

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

lβ = −β β + 1 = ( . b∗

1b∗ 2 · · · )−β

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 endpoint rβ =

1 β+1:

rβ = 1 β + 1 = ( . 0b∗

1b∗ 2b∗ 3 · · · )−β,

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 3 ∈ I−β.

(−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

• dd period q, i.e., b∗

i+q = b∗ i for all i ≥ 1, then we define

c∗

i =

     i ≡ 1 mod (q + 1) b∗

i−1 mod (q+1)

i ≡ 0, 1 mod (q + 1), b∗

q+1 − 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 X3 − 2X2 + 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≥sn

1 (−β)n,

where s0 = lβ, and si = T−β(si−1). Then the measure

dµ = h−βdλ 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. (β3 = β + 1)

▽(−β)-expansion of real numbers – p.22

Example

Let β be the minimal Pisot number. (β3 = β + 1)

lβ = s0 < s3 < s1 < s2, s3 = s4 = · · · .

▽(−β)-expansion of real numbers – p.22

Example

Let β be the minimal Pisot number. (β3 = 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 1 β2

▽(−β)-expansion of real numbers – p.22

Example

Let β be the minimal Pisot number. (β3 = 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 1 β2

µ is not equivalent to

the Lebesgue mea- sure.

▽(−β)-expansion of real numbers – p.22

Example

Let β be the minimal Pisot number. (β3 = 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 1 β2

µ is not equivalent to

the Lebesgue mea- sure.

(−β)-expansion of real numbers – p.22

Thank you very much.

(−β)-expansion of real numbers – p.23