Measures, Dimension, and Ergodic Theory

University Of Washington, Seattle WA

Investigators

Abstract

DMS 0355187 B Solomyak University of Washington MEASURES, DIMENSION, AND ERGODIC THEORY ABSTRACT This project is in the area of interaction between geometric measure theory, fractal geometry, and ergodic theory. Iterated function systems, popularly known as the ``Chaos Game,'' provide a convenient framework for the study of fractal phenomena. Invariant measures for such systems include infinite sums of products of idependent, identically-distributed random variables, which arise in many probablistic and dynamical models, and stationary measures for random matrix products. One of the central problems is to determine when such a measure is absolutely continuous. A closely related line of research is concerned with topological properties of attractors, in particular, self-similar and self-affine sets. Especially challenging are the systems that are not uniformly contracting and those which have a substantial ``overlap.'' On the ergodic theory side, the project deals with substitution and tiling dynamical systems without discrete spectrum, as well as with algebraic coding of toral automorphisms. The so-called non-Pisot case is of particular interest to us, where it is proposed to use some variants of the beta-transformation defined on self-affine sets. To play the ``Chaos Game'' on the plane, one should specify a family of planar tranformations (called an iterated function system) and iteratively apply one of them at random, with prescribed probabilities. Under certain technical conditions (called ``contracting-on-average'') the emerging picture will almost surely ``converge'' to a set called the attractor of the iterated function system. This is a popular method to draw fractals on the computer screen, but it is more than a game: iterated functions are widely used in signal processing, image compression, and simulation algorithms, as well as in mathematical dynamical systems theory. The picture that we see on a computer screen is actually a greytone image, which is an approximation of a measure, or probability distribution. An important problem is to decide when this distribution has a density. A related line of research is to classify the ``zoo'' of fractal objects which arise in the course of the Chaos Game. The second part of the project is concerned with another class of objects which can be obtained by iteration: infinite substitutive sequences and self-similar tilings. Now the iteration procedure involves replacing each symbol by a block of symbols, or each tile by a patch of tiles. The limiting object is then used to create a fascinating dynamical system which we study. Apart from their intrinsic beauty, such systems have found applications in physics.