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(−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