Ergodic Theory, Dynamics and Fractals

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In this project, the Principal Investigator will build on his earlier work on substitutions, tiling dynamical systems, Bernoulli convolutions, and related topics, to further advance basic mathematical knowledge. Specific goals include extending the orbit equivalence theory and spectral theory to nonminimal Bratteli-Vershik and substitution systems, characterizing invariant measures for nonprimitive substitution tilings, and developing multifractal analysis of Bernoulli convolutions and related self-affine measures. It is expected that new links and connections between diverse areas will emerge in the course of this investigation. In particular, flows along stable and unstable foliations of pseudo-Anosov maps are linked to substitution systems, nonprimitive tilings give rise to fractals, such as the Sierpinski gasket self-similar tiling and the familiar fractal Sierpinski gasket set, complex rational (e.g., quadratic) and piecewise linear dynamics are sometimes related by a conjugacy. The methods and techniques used will not be limited to those from ergodic theory and dynamics but will also come from other fields, including operator algebras, combinatorics, number theory, probability, and complex analysis.The fields of ergodic theory, dynamical systems, and fractal geometry have their origins, motivation, and applications in many fields outside of mathematics, including statistical and celestial mechanics, population biology, aero- and fluid-dynamics, and materials science. A dynamical system is given by a so-called phase space and some transformation rules that prescribe how states change in time. Here "time" may be discrete or continuous, multidimensional, or more generally, given by a group. Dynamics may be regular or chaotic, or it could exhibit a "mixed" behavior. Many dynamically defined objects turn out to be "fractals." Mathematicians often study simplified models that capture the important features of the physical system in the simplest possible form. This project aims to analyze in depth several existing models, as well as develop new ones. Although it is primarily theoretical in its focus, it will advance basic knowledge in areas with diverse connections and applications. Specifically, Bernoulli convolutions and related fractal constructions have been used in signal and image processing, control theory, and econometrics; substitutions and tilings have many links to computer science and solid-state physics. The PI interacts with scientists and participates in professional meetings of an interdisciplinary nature. In addition, this project will contribute to the development of human resources through training undergraduate and graduate students.