The DYNAMICS & GEOMETRY of MULTIRESOLUTION METHODS Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore Tel (65) Fax (65)

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 AD Sumerian and Babylonian (Akkadian) BC (I, V, X, L, C, D) = (1,5,10,50,100,500)

INTEGER REPRESENTATION Integers Digits andDefine functions Base and digit sequences Theorem An integer n admits an (m,D)-expansion if and only if the trajectory of n under f converges to 0 Proof

INTEGRAL DYNAMICS is a basin of attraction since For the dynamical system and for everythere exists a positive integer such that Therefore, the orbit of every point converges to a periodic orbit contained in S.

EXAMPLES all nonnegative integers admit expansions all integers admit expansions some nonnegative integers admit expansions

RELATED PROBLEM 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 Theorem Fractions Proof since 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 is compact, and

REAL NUMBER REPRESENTATION satisfies refinement equation If then wheresatisfies and

REAL NUMBER REPRESENTATION Proof (If) Theorem 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 by Then observe that However, uniqueness occurs if and only if

PROJECTIVE GEOMETRY AND NONNEGATIVE MATRICES Proof TheoremPappus (conjectured 500 BC, proved 300 AD) Cross ratios are invariant under projective maps CorollaryPerron-Frobenius theorem for nonnegative matrices

RUELLE TRANSFER OPERATORS TheoremIfand then Theorem CorollaryUniqueness if and only if D is relatively prime and nice then If