Topics in Fractal Geometry, Dynamics, and Ergodic Theory

University Of Washington, Seattle WA

Investigators

Abstract

This project addresses problems in dimension theory of dynamical systems and ergodic theory of tilings and substitutions. In the first direction, it is proposed to study the dimension characteristics of invariant sets for non-conformal dynamical systems and natural measures on them. These questions are related to problems on self-similar sets with overlap and arithmetic sums of Cantor sets. The second direction concerns non-periodic self-affine tilings, their generalizations, and symbolic analogs. There are many links between the two directions; for instance, self-affine tiles often have fractal boundaries, and number-theoretic issues (e.g., the appearance of PV-numbers) are prominent in both areas. The problems to be studied are concerned with dynamical and diffraction spectra of tilings and substitutions. The mathematical chaos theory deals with the appearance of chaotic motions in dynamical systems (which are mathematical models of various phenomena in physics, biology, economics, etc.). These chaotic motions are often associated with various "fractal" phenomena. The modern dimension theory uses a wide variety of tools to study the fine structure of these motions and their invariant sets (such as "strange attractors"). Still, many aspects of this "fractal chaotic world" remain mysterious, and the proposed research aims to get further insights into them. The second part of the project is concerned with the so-called "aperiodic order," the hallmark of which is the remarkable class of solids, discovered in the early 1980's and called quasicrystals, which exhibit sharp diffraction spectrum with a forbidden 10-fold symmetry. Penrose tilings, discovered in the 1970's in a purely mathematical development, turned out to be a good model for certain quasicrystals, but many open problems remain, both on the mathematical and on the physical side. The second part of the proposed research investigates some of these problems using the techniques of ergodic theory and symbolic dynamics.