Dimension and Dynamics

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ABSTRACTSolomyak The goal of this project is to investigate self-similar measures and sets, self-affine sets, and attractors of non-linear iterated function systems. The main emphasis is on the "overlapping" case when the symbolic coding is non-unique, and the closely related higher-dimensional non-conformal case. Hyperbolic and parabolic systems will be considered. The topics includedimension computation, the study of topological properties of attractors, and smoothness properties of natural measures on them. The methods of symbolic dynamics and ergodic theory, number theory and geometric measure theory, harmonic and complex analysis will be used. Dynamical systems describe a wide range of phenomena,such as the motion of planets and asteroids, the rise and fall of populations, the weather, and the stock market. Many of these systems exhibit "chaotic" behavior and have "attractors" of intricate shapes and structure. This structure can often be explained by an underlying "self-similarity" which means thatthe object looks similar at all scales. The "fractal dimension" of the attractor is one of the basic characteristics which can be used to distinguish systems of different nature.Topological properties, such as connectedness and presence of the interior are also very important. We are still veryfar from complete understanding of these complex objects, and this study aims to investigate them in a number of model cases. Moreover, the theory developed for this is relevant inother areas, such as computer image recognition.