Fast arithmetical algorithms in M obius number systems Petr K - PowerPoint PPT Presentation

Fast arithmetical algorithms in M¨ obius number systems Petr K˚ urka Center for Theoretical Study Academy of Sciences and Charles University in Prague Valparaiso, November 2011

Iterative systems X compact metric space, A finite alphabet ( F a : X → X ) a ∈ A continuous. ( F u : X → X ) u ∈ A ∗ , F uv = F u ◦ F v , F λ = Id Theorem(Barnsley) If ( F a : X → X ) a ∈ A are contractions, then there exists a unique attractor Y ⊆ X with Y = � a ∈ A F a ( Y ), and a continuous surjective symbolic mapping Φ : A N → Y � F u [0 , n ) ( X ) , u ∈ A N { Φ( u ) } = n > 0

Binary system A = { 0 , 1 } , Φ 2 : A N → [0 , 1] F 0 ( x ) = x 2 , F 1 ( x ) = x + 1 2 � u i · 2 − i − 1 , u ∈ A N Φ 2 ( u ) = i ≥ 0 [1] [0] 0 1 a → F − 1 Expansion graph: x a ( x ) if x ∈ W a = F a [0 , 1]

Binary signed system is redundant A = { 1 , 0 , 1 } , Φ 3 : A N → [ − 1 , 1] F 1 ( x ) = x − 1 2 , F 1 ( x ) = x + 1 , F 0 ( x ) = x 2 2 � u i · 2 − i − 1 , u ∈ A N Φ 3 ( u ) = i ≥ 0 [1] [0] - [1] -1 1

Real orientation-preserving M¨ obius transformations M a , b , c , d ( x ) = ax + b cx + d , ad − bc > 0 act on R = R ∪ {∞} and on U = { z ∈ C : ℑ ( z ) > 0 } Stereographic projection d ( z ) = iz +1 z + i d : R → ∂ D = { z ∈ C : | z | = 1 } unit circle d : U → D = { z ∈ C : | z | < 1 } unit disc

obius transformations � M = d ◦ M ◦ d − 1 Disc M¨ 1/0 1/0 1/0 -4 4 -3 3 -2 -2 2 2 -1 1 -1 1 -1 1 -1/2 1/2 -1/4 1/4 0 0 0 � F 1 ( z ) = (2+ i ) z +1 � � (7+2 i ) z + i F 0 ( z ) = 3 z − i F 2 ( z ) = iz +3 z +2 − i − iz +(7 − 2 i ) F 2 ( x ) = 4 x +1 F 0 ( x ) = x / 2 F 1 ( x ) = x + 1 3 − x hyperbolic parabolic elliptic � Mean value E ( � z d ( � M ℓ ) = � M ℓ ) = M (0) ∂ D

Circle metric and derivation the length of arc between d ( x ) and d ( y ): | x − y | ̺ ( x , y ) = 2 arcsin � ( x 2 + 1)( y 2 + 1) circle derivation of M ( x ) = ( ax + b ) / ( cx + d ): ̺ ( M ( x ) , M ( y )) M • ( x ) = lim ̺ ( x , y ) y → x ( ad − bc )( x 2 + 1) = ( ax + b ) 2 + ( cx + d ) 2

Contracting and expanding intervals U u = { x ∈ R : F • u ( x ) < 1 } , F u ( U u ) = V u V u = { x ∈ R : ( F − 1 u ) • ( x ) > 1 } ∞ √ √ − 2 2 V √ √ 2 2 − 2 2 − 1 1 V 2 2 U U F(x)=x/2 F(x)=x+1

M¨ obius number system(MNS) ( F , W ) ( F a : R → R ) a ∈ A M¨ obius transformations W a ⊆ V a expansion intervals � a ∈ A W a = R a → F − 1 Expansion graph: x a ( x ) if x ∈ W a u 0 u 1 u 2 → F − 1 → F − 1 u 0 ( x ) u 0 u 1 ( x ) → · · · x x ∈ W u 0 , F − 1 u 0 ( x ) ∈ W u 1 , F − 1 u 0 u 1 ( x ) ∈ W u 2 W u := W u 0 ∩ F u 0 ( W u 1 ) ∩ · · · ∩ F u [0 , n ) ( W u n ) x ∈ W u iff u is the label of a path with source x :

Expansion subshift: S W := { u ∈ A N : ∀ n , W u [0 , n ) � = ∅} Symbolic extension: Φ( u ) = lim n →∞ F u [0 , n ) ( i ) ∈ R , u ∈ S W Φ : S W → R is continuous and surjective.

1 = F a 0 1 F 0 F a 1 Continued fractions a 0 − 1 F 0 · · · 1 a 1 − a 2 − · · · 1/0 1/0 F 1 ( x ) = x − 1 , W 1 = ( ∞ , − 1) -4/1 -4/1 4/1 4/1 -3/1 -3/1 3/1 3/1 F 0 ( x ) = − 1 / x , W 0 = ( − 1 , 1) -2/1 -2/1 111 111 --- 2/1 2/1 F 1 ( x ) = x + 1 , W 1 = (1 , ∞ ) -- 11 11 110 110 -3/2 -3/2 -- 3/2 3/2 - 101 - 101 - 10 10 S W is a SFT. 1 1 - forbidden words: 0 -1/1 -1/1 1/1 1/1 - 00 , 11 , 11 , 101 , 101 01 01 0 0 1 1 0 0 - -- 011 011 a → F − 1 a ( x ) if x ∈ W a -1/2 -1/2 x 1/2 1/2 - -1/3 -1/3 1/3 1/3 1 1 1 -2 -1 0 1 2 3 0/1 0/1 0

Binary signed system, A = { 1 , 1 , 2 } -8/1 -8/1 F 1 ( x ) = ( x − 1) / 2 8/1 8/1 -4/1 -4/1 4/1 4/1 F 1 ( x ) = ( x + 1) / 2 -2/1 -2/1 F 2 ( x ) = 2 x 222 2/1 2/1 22 221 221 - 1 -3/2 -3/2 3/2 3/2 W 1 = ( − 1 , 0) → ( − 1 , 1) 211 211 -- 1 2 W 1 = (0 , 1) → ( − 1 , 1) 21 21 - 2 → ( 1 2 , − 1 W 2 = (1 , − 1) 2 ) -1/1 -1/1 1/1 1/1 111 111 -7/8 -7/8 7/8 7/8 --- 1 1 1 1 1 1 - S W is a SFT. - - -3/4 -3/4 3/4 3/4 - 111 111 forbidden words: -- -5/8 -5/8 5/8 5/8 12 , 12 , 211 , 211 - 11 11 - -1/2 -1/2 1/2 1/2 1 1 - - 1 1 1 1 - -3/8 -3/8 3/8 3/8 S W = { 2 n u : u ∈ { 1 , 1 } N } 111 -- - 111 Φ(2 n u ) = � ∞ -1/4 -1/4 1/4 1/4 i =0 u i · 2 n − i -1/8 -1/8 1/8 1/8 0/1 0/1

Fractional bilinear functions P ( x , y ) = axy + bx + cy + d exy + fx + gy + h , M ( x ) = ax + b cx + d .     a 0 b 0 a b 0 0     0 a 0 b c d 0 0 M x =  , M y =        c 0 d 0 0 0 a b 0 c 0 d 0 0 c d P ( Mx , y ) = PM x ( x , y ), P ( x , My ) = PM y ( x , y ), MP ( x , y ) are fractional bilinear functions.

Bilinear graph vertices: ( P , u , v ), u , v ∈ S W . a → ( F − 1 P ( W u 0 , W v 0 ) ⊆ W a ( P , u , v ) a P , u , v ) if λ → ( PF x ( P , u , v ) u 0 , σ ( u ) , v ) λ → ( PF y ( P , u , v ) v 0 , u , σ ( v )) Proposition If u , v ∈ S W and w ∈ A N is a label of a path with source ( P , u , v ), then w ∈ S W and Φ( w ) = P (Φ( u ) , Φ( v )).

Linear graph vertices: ( M , u ) ∈ M 1 × S W , a → ( F − 1 emission: ( M , u ) a M , u ) if M ( W u 0 ) ⊆ W a λ absorption: ( M , u ) → ( MF u 0 , σ ( u )) Proposition There exists a path with source ( M , u ) whose label w = f ( u ) ∈ S W and Φ( w ) = M (Φ( u )). If M remain bounded, then the algorithm has linear time complexity.

Linear graph vertices: ( M , u ) ∈ M 1 × S W , a → ( F − 1 emission: ( M , u ) a M , u ) if M ( W u 0 ) ⊆ W a λ absorption: ( M , u ) → ( MF u 0 , σ ( u )) Proposition There exists a path with source ( M , u ) whose label w = f ( u ) ∈ S W and Φ( w ) = M (Φ( u )). If M remain bounded, then the algorithm has linear time complexity.

Bimodular group: det( M ) = 2 p - 2101 - 2011 - 1102 - 1012 1012 1102 2011 2101 - 210 - 201 - 110 - 101 101 110 201 210 3 - 21 20 - 11 10 10 11 20 21 2 2 1 1 1 1 2 2 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 -1 __ -3 __ -2 __ -1 __ -1 __ -1 __ -1 __ 0 _ 0 _ 1 _ _ 1 _ 1 1 _ _ 2 3 _ _ 1 0 1 1 1 1 2 3 1 1 3 2 1 1 1 1 0 det( M ) = 2 , || M || = 6 , tr ( M ) = 3 0 1 2 3 a x +1 2 x x 2 x + 1 F a x +2 2 x +1 ( − 1 ( 1 W a = V a 3 , 1) (0 , 2) 2 , ∞ ) (1 , − 3)

Redundant bimodular system W a = V ( F a ) -7/1 1/0 -5/1 7/1 5/1 1 / 1 3 / - 3 1 / 1 4 3 2 4 3 / - 2 45 32 34 43 46 53 24 31 52 25 -3/2 3/2 4 35 42 3 -4/3 26 51 4/3 54 47 30 23 64 5 2 13 5 2 5 2 65 12 -1/1 40 73 04 37 1/1 56 21 6 1 6 6 57 20 1 1 -3/4 3/4 62 15 74 03 -2/3 2/3 67 06 7 0 71 10 60 75 02 17 61 16 - 1 1 07 70 / / 2 7 0 2 76 7 0 01 - 1 1 / / 3 3 -1/5 1/5 -1/7 1/7 0/1 Top five algorithm: Keeps five matrices with the smallest norm.

Ergodic theory of singular transformations M ( x ) = ax + b cx + d , ad − bc = 0 2 : ( ax 0 + bx 1 ) y 1 = ( cx 0 + dx 1 ) y 0 } � M = { ( x , y ) ∈ R = ( R × { s M } ) ∪ ( { u m } × R ) s M = M ( i ) ∈ { a c , b d } ∩ R : stable point u M = M − 1 ( i ) ∈ {− b a , − d c } ∩ R : unstable point If M is singular, then s MF = s M , u FM = u M . Emission acts on columns, absorption acts on rows of singular matrices.

� x 2 0 + x 2 Growth of norm || x || = 1 ( ad − bc )( x 2 0 + x 2 1 ) M • ( x 0 x 1 ) = ( ax 0 + bx 1 ) 2 + ( cx 0 + dx 1 ) 2 = det( M ) · || x || 2 || M ( x )) || 2 � || M ( x ) || det( M ) = M • ( x ) || x ||

Invariant emission measure Partition of unity w a : R → [0 , 1], a ∈ A � supp ( w a ) ⊆ W a , w a ( x ) = 1 a ∈ A w a → F − 1 Emission process x a ( x ) has a unique Lebesgue-continuous invariant measure µ . � � 1 ln det( F u ) e n = u ) • d µ ( F − 1 2 u ∈L n ( S W ) e n + m ≤ e n + e m : emission quotients

Invariant absorption measure � � w a · w b ◦ F − 1 P a = w a d µ, P ab = d µ a S W is a SFT of order 2: R ab = P ab / P a . R ab → ( F t Absorption process ( x , a ) − b ( x ) , b ) in R × A has a unique invariant measure ν ( U , a ) = P a ν a ( U ) � � 1 ln det( F u ) a n = u ) • d ν a 2 P au ( F t au ∈L n ( S W ) a n + m ≤ a n + a m : absorption quotients

Transaction quotient E = n →∞ exp( e n / n ) : lim Emission quotient A = n →∞ exp( a n / n ) : lim Absorption quotient T = E · A : Transaction quotient T > √ r : positional r -ary system (Heckmann). T < 1 . 1: redundant bimodular system.

Conjecture There exists a multiplication algorithm with average linear time complexity.