The DYNAMICS & GEOMETRY of MULTIRESOLUTION METHODS

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email wlawton@math.nus.edu.sgTel (65) 874-2749Fax (65) 779-5452

OUTLINE

Dynamical Systems and Positional Notation

Multiresolution Computational Methods

Projective Geometry and Nonnegative Matrices

Transfer Operators in Statistical Physics and QFT

Refinable Functions and Wavelets

Recent Progress

POSITIONAL NOTATION - HISTORY

“The invention of positional notation was the first profound mathematical advance. It made accurate and efficient calculations possible”, Mathematics in Civilization, H. L. Resnikoff & R. O. Wells, Jr.,Dover, NY,1973,1984

Romans 750 BC - 476 AD

Sumerian and Babylonian (Akkadian) 6000-0 BC

}59,...,2,1,0{,...d,c,b,a ...60/dcb60a3600,...d;c,b,a

(I, V, X, L, C, D) = (1,5,10,50,100,500)

Z

DZ:h

INTEGER REPRESENTATION

Integers

Digits

andDefine functions

Base }1,0,1{\Zm

DmZZ|,m||D|,ZD0

ZZ:f Zn),n(fm)n(hn

and digit sequences

)n(fmnmmnn 1k1kk

k10

Zn,0j)),n(f(hn jj

Theorem An integer n admits an (m,D)-expansion if and only if the trajectory of n under f converges to 0

Proof )n(fmmnn)n(mfnn 22100

}1|m|

}Dd|:dmax{||n|:n{S

INTEGRAL DYNAMICS

is a basin of attraction since

ZZ:f For the dynamical system

S)S(f

and for every Zn there exists a positive integer

)n(k such that S)n(f )n(k

Therefore, the orbit of every point converges to a periodic orbit contained in S.

EXAMPLES

all nonnegative integers admit expansions

}1,0,1{S},9....,1,0{D,10m }0{})1,0({f},1{})1({f

}0{S},5,...,3,4{D,10m all integers admit expansions

}1,0,1{S},1,0{D,2m }0{})1,0({f},1{})1({f

all integers admit expansions

}2,1,0,1,2{S},5,1,0{D,3m }2{})1({f},1{})2,2({f},0{})1,0({f

some nonnegative integers admit expansions

RELATED PROBLEM

even,oddnfor2/n,2/)1n3(:)n(T

L. Collatz 1932 introduced the function

and conjectured that all its trajectories converge to {1,2}

Known as Ulam’s problem in computer science, it has connections with undecidability, numerical analysis, number theory and probability

REAL NUMBER REPRESENTATION

Dd:dm:F j1j

jj

Theorem FZR

Fractions

Proof

since FZ)FmZD(m)FZ(m 11 FZZm j

0j

FZ is closed since

Corollary F has nonempty interior and measure at least one, with measure one if and only if the representation is unique almost everywhere

it contains the dense set

F is compact, and

REAL NUMBER REPRESENTATION

satisfies refinement equation

0else,Dn if1)n(c If

then

h)h( W where

j )jmx()j(c)x(

RZ:c

satisfies

and

0else,Fx if1)x( RR:

RZ:h dx)nx()x()n(h

)Z(R)Z(R: 00 WL]}[-L,supp(f):RZ:f{:)Z(R0

)mk(fcc~m)k)(f( 1 W

)1|m/(|))Dmin()D(max(:L

REAL NUMBER REPRESENTATION

Proof (If)

Theorem

)(W

Construct the sequence

Almost all real numbers have unique representation (Z + F) if and only if 1 is a simple eigenvalue of W

is represented by a nonnegative matrix W

h

)Z(R0

W

0else,0n if1)n( by

Then observe that

However, uniqueness occurs if and only if

PROJECTIVE GEOMETRY AND NONNEGATIVE MATRICES

Proof

Theorem Pappus (conjectured 500 BC, proved 300 AD)

Cross ratios are invariant under projective maps

Corollary Perron-Frobenius theorem for nonnegative matrices

RUELLE TRANSFER OPERATORS

XX:A }0{RX:r

)X(R)X(R: R

x)y(A

)y(f)y(r)x)(f(R

Theorem If h)h( R0h and

then ))h(set_zero(A)h(set_zero Theorem

RFFWZ/RX

j j

j ))xid2exp((m)x(rCorollary Uniqueness if and only if D is relatively prime

and nice

Zmodmx)x(A then

If