Topics in Fractal Geometry, Dynamics, and Ergodic Theory

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This project addresses problems in dimension theory ofdynamical systems and ergodic theory of tilings and substitutions.In the first direction, it is proposed to study the dimensioncharacteristics of invariant sets for non-conformal dynamicalsystems and natural measures on them. These questions are relatedto problems on self-similar sets with overlap and arithmetic sumsof Cantor sets. The second direction concerns non-periodic self-affinetilings, their generalizations, and symbolic analogs. There are manylinks between the two directions; for instance, self-affine tiles oftenhave 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 diffractionspectra of tilings and substitutions.The mathematical chaos theory deals with the appearance of chaoticmotions in dynamical systems (which are mathematical models ofvarious phenomena in physics, biology, economics, etc.). Thesechaotic motions are often associated with various "fractal" phenomena.The modern dimension theory uses a wide variety of tools tostudy 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 researchaims to get further insights into them. The second part of the projectis concerned with the so-called "aperiodic order," the hallmarkof which is the remarkable class of solids, discovered in theearly 1980's and called quasicrystals, which exhibit sharp diffractionspectrum with a forbidden 10-fold symmetry. Penrose tilings,discovered in the 1970's in a purely mathematical development, turnedout to be a good model for certain quasicrystals, but many open problemsremain, both on the mathematical and on the physical side.The second part of the proposed research investigates some of theseproblems using the techniques of ergodic theory and symbolic dynamics.