Fractals and Tilings

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This project is devoted to several problems in fractal geometry and the theory of tilings and substitutions. Bernoulli convolutions are, perhaps, the simplest examples of self-similar measures with overlaps. They appeared in many different branches of mathematics and science, including signal processing, number theory and harmonic analysis.This project will investigate novel phenomena related to their multifractal spectrum, which are linked to expansions in non-integer bases. It will also address random analogs of Bernoulli convolutions, such as "branching random walks with exponentially decreasing steps."In the area of tilings and tiling dynamical systems, it is proposed to complete the characterization of expansion maps for self-affine tilings, which should lead to advances in problems on associated dynamical systems. This program was started by Thurston and Kenyon, and it has connections with geometric dynamics, theory of laminations, and rigidity theory.Another question which will be addressed is about the nature of the continuous spectral component of substitution dynamical systems.Many of the sets which appear in modern mathematics have complicated structure and cannot be described analytically. Mandelbrot introduced the term "fractal" for such sets and demonstrated that they can be useful for modeling many natural phenomena. On the other hand, mathematicians such as Peano, Cantor and Weierstrass, investigated mathematical fractals (without using this term) since the 19th century.The simplest and most basic among fractal objects are self-similar sets and measures. "Self-similarity" means, roughly speaking, that "zooming in"into the object one can see similar pictures at arbitrarily small scales.Also important are fractalsdefined probabilistically, which exhibit statistical self-similarity; they often serve as more realistic models in applications. This project will address several delicate questions on the structure of fractal sets.The second part of the proposal has to do with tilings and substitutions; it has links to fractals, but also to combinatorics and discrete geometry.Self-similar tilings, such as the Penrose tilings, have been used as models for quasicrystals, and the associated dynamical systems and their spectral properties turned out to be relevant for solid-state physics.