( ) -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 − k β k + x − k +1 β k − 1 + · · · + x 0 + x 1 β + x 2 β 2 + · · · , where k ≥ 0 is a certain integer and x i > 0 for i ≥ − k . It is denoted by x = ( x − k x − k +1 · · · x 0 . x 1 x 2 · · · ) β . ( − β ) -expansion of real numbers – p.1

Review of β -expansions The β - transformation T β : [0 , 1) → [0 , 1) is defined by T β ( 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 . x 1 x 2 · · · ) β . where x i = ⌊ βT i − 1 ( x ) ⌋ for i ≥ 1 . β We call this representation the β -expansion of x . ( − β ) -expansion of real numbers – p.3

Review of β -expansions A sequence ( x 1 , x 2 , . . . ) is admissible if there exists x ∈ [0 , 1) such that x = (0 . x 1 x 2 . . . ) β is the β -expansion of x . ▽ ( − β ) -expansion of real numbers – p.4

Review of β -expansions A sequence ( x 1 , x 2 , . . . ) is admissible if there exists x ∈ [0 , 1) such that x = (0 . x 1 x 2 . . . ) β is the β -expansion of x . Theorem 2 (Parry) . A sequence ( x 1 , x 2 , . . . ) is admissible if and only if ( x 1 , x 2 , . . . ) ≺ 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 . d 1 d 2 . . . ) . Then we have a β -representation of 1 : 1 = (0 . ⌊ β ⌋ d 1 d 2 · · · ) β . � ( ⌊ β ⌋ , d 1 , d 2 , . . . , d i − 1 , d i − 1) 0 = d i +1 = d i +2 = · · · d ∗ (1 , β ) := otherwise ( ⌊ β ⌋ , d 1 , d 2 , . . . ) ( − 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 h β : [0 , 1) → R be defined by 1 � h β ( x ) = β n , x ≤ s n where s 0 = 1 and s n = T n − 1 ( { β } ) for n ≥ 1 . Then the measure β dµ = h β dx is invariant under T β 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 ( x 1 , x 2 , . . . ) ≺ 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 ( x 1 , x 2 , . . . ) ≺ lex d ∗ (1 , β ) , ∀ i ≥ 1 . The value of β -transformation can be expressed as, T β ( 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 ( x 1 , x 2 , . . . ) ≺ lex d ∗ (1 , β ) , ∀ i ≥ 1 . The value of β -transformation can be expressed as, T β ( 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 ( x 1 , x 2 , . . . ) ≺ lex d ∗ (1 , β ) , ∀ i ≥ 1 . The value of β -transformation can be expressed as, T β ( 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 ( x 1 , x 2 , . . . ) ≺ lex d ∗ (1 , β ) , ∀ i ≥ 1 . The value of β -transformation can be expressed as, T β ( 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 ( x 1 , x 2 , . . . ) ≺ lex d ∗ (1 , β ) , ∀ i ≥ 1 . The value of β -transformation can be expressed as, T β ( x ) = { βx } = { βx − 0 } +0 0 is the left endpoint of [0 , 1) . x i = ⌊ 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 + · · · + x 0 + x 1 x 2 ( − β )+ ( − β ) 2 + · · · , where k ≥ 0 is a certain integer and x i > 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 + · · · + x 0 + x 1 x 2 ( − β )+ ( − β ) 2 + · · · , where k ≥ 0 is a certain integer and x i > 0 for i ≥ − k . It is denoted by x = ( x − k x − k +1 · · · x 0 . x 1 x 2 · · · ) − β . ( − β ) -expansion of real numbers – p.8

Definition: ( − β ) -transformation � � − β 1 I β = [ l β , r β ) = β +1 , . β +1 ▽ ( − β ) -expansion of real numbers – p.9

Definition: ( − β ) -transformation � � − β 1 I β = [ l β , r β ) = β +1 , . β +1 The ( − β ) -transformation T − β on I β is defined by T − β ( x ) = {− βx − l β } + l β ▽ ( − β ) -expansion of real numbers – p.9

Definition: ( − β ) -transformation � � − β 1 I β = [ l β , r β ) = β +1 , . β +1 The ( − β ) -transformation T − β on I β is defined by T − β ( x ) = {− βx − l β } + l β � β � = − βx − − βx + β + 1 ( − β ) -expansion of real numbers – p.9

Definition β = 2 . 3 1 β +1 β 1 − β +1 β +1 β − β +1 ▽ ( − β ) -expansion of real numbers – p.10

Definition β = 2 . 3 1 β +1 β 1 − β +1 β +1 β − β +1 ▽ ( − β ) -expansion of real numbers – p.10

Definition β = 2 . 3 1 β +1 2 1 0 β 1 − β +1 β +1 β − β +1 ( − β ) -expansion of real numbers – p.10

Definition Then, for each x ∈ I β , we have a particular ( − β ) -representation x = ( . x 1 x 2 · · · ) − β . where x i = ⌊− βT i − 1 − β ( x ) − l β ⌋ 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 I β , there is an integer d such that x/ ( − β ) d ∈ I β , hence we have the ( − β ) -expansion of x : x = ( x − d +1 x − d +2 · · · x 0 . x 1 x 2 · · · ) − β (1) where x − d + i = ⌊− βT i − 1 β x β +1 ⌋ . − β ( ( − β ) d ) + ( − 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 , − 1 = (11 . ) − 2 , 3 = (111 . ) − 2 , − 2 = (10 . ) − 2 , 4 = (100 . ) − 2 , − 3 = (1101 . ) − 2 , . . . . . . 100 = (110100100 . ) − 2 , − 100 = (11101100 . ) − 2 . . . . . . ▽ ( − β ) -expansion of real numbers – p.13

Examples Example 4. β = 2 2 = (110 . ) − 2 , − 1 = (11 . ) − 2 , 3 = (111 . ) − 2 , − 2 = (10 . ) − 2 , 4 = (100 . ) − 2 , − 3 = (1101 . ) − 2 , . . . . . . 100 = (110100100 . ) − 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 , − 1 = (11 . ) − 2 , 3 = (111 . ) − 2 , − 2 = (10 . ) − 2 , 4 = (100 . ) − 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 , − 1 = (11 . ) − 2 , 3 = (111 . ) − 2 , − 2 = (10 . ) − 2 , 4 = (100 . ) − 2 , − 3 = (1101 . ) − 2 , . . . . . . 100 = (110100100 . ) − 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 , − 1 = (11 . 001) − β , 3 = (100 . 111001) − β , − 2 = (10 . 001) − β , 4 = (101 . 111001) − β , . . . . . . − 100 = (1100010010 . 01000100000100 100 = (111000110 . 00001100101111) − β , . . . . . . ( − β ) -expansion of real numbers – p.14

Admissible sequences We say an integer sequence ( x 1 , x 2 , . . . ) is ( − β ) -admissible , if there exists a real number x ∈ I β such that x = ( . x 1 x 2 · · · ) − β is a ( − β ) -expansion. ▽ ( − β ) -expansion of real numbers – p.15