The DYNAMICS & GEOMETRY of MULTIRESOLUTION METHODS

1. 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.sg Tel (65) 874-2749 Fax (65) 779-5452

2. OUTLINE Dynamical Systems and Positional Notation Projective Geometry and Nonnegative Matrices Transfer Operators in Statistical Physics and QFT Multiresolution Computational Methods Refinable Functions and Wavelets Recent Progress

3. POSITIONAL NOTATION - HISTORY Romans 750 BC - 476 AD (I, V, X, L, C, D) = (1,5,10,50,100,500) Sumerian and Babylonian (Akkadian) 6000-0 BC “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

4. INTEGER REPRESENTATION Integers Base Digits Define functions and 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

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

6. EXAMPLES all nonnegative integers admit expansions all integers admit expansions all integers admit expansions some nonnegative integers admit expansions

7. 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

8. REAL NUMBER REPRESENTATION Fractions Theorem Proof is closed since is compact, and it contains the dense set since Corollary F has nonempty interior and measure at least one, with measure one if and only if the representation is unique almost everywhere

9. REAL NUMBER REPRESENTATION If then satisfies refinementequation and satisfies where

10. REAL NUMBER REPRESENTATION Theorem is represented by a nonnegative matrix W Almost all real numbers have unique representation (Z + F) if and only if 1 is a simple eigenvalue of W Proof (If) Construct the sequence by Then observe that However, uniqueness occurs if and only if

11. PROJECTIVE GEOMETRY AND NONNEGATIVE MATRICES Theorem Pappus (conjectured 500 BC, proved 300 AD) Proof Cross ratios are invariant under projective maps Corollary Perron-Frobenius theorem for nonnegative matrices

12. RUELLE TRANSFER OPERATORS and nice Theorem If and then Theorem If then Corollary Uniqueness if and only if D is relatively prime