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This volume contains the proceedings of the virtual AMS Special Session on Fractal Geometry and Dynamical Systems, held from May 14-15, 2022. The content covers a wide range of topics. It includes nonautonomous dynamics of complex polynomials, theory and applications of polymorphisms, topological and geometric problems related to dynamical systems, and also covers fractal dimensions, including the Hausdorff dimension of fractal interpolation functions. Furthermore, the book contains a discussion of self-similar measures as well as the theory of IFS measures associated with Bratteli diagrams. This book is suitable for graduate students interested in fractal theory, researchers interested in fractal geometry and dynamical systems, and anyone interested in the application of fractals in science and engineering. This book also offers a valuable resource for researchers working on applications of fractals in different fields.
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Articles in this volume are based on talks given at the special session ""Dynamical Systems: Statistical Properties, Spectral Theory, and Fractal Geometry"" at the AMS Sectional Meeting held at University of Texas at San Antonio on September 14-15, 2024. The authors discuss topics at the intersection of dynamical systems, ergodic theory, fractal geometry, operator theory, and symbolic dynamics. Key features include the introduction of relative topological pressure for nonautonomous systems, a novel dual-operator framework for analyzing diffusion equations with self-similar measures, and generalized fractal interpolation functions. Several articles offer new methods that extend classical theories to more intricate contexts. Readers will benefit from a wide range of fresh perspectives, including asymptotic complexity in billiards, ergodic Hilbert transform inequalities, symbolic zeta function poles, and new transformations in measurable dynamics. The unique blend of topological, combinatorial, probabilistic, and analytical tools offers a valuable resource for researchers seeking to explore or expand contemporary methods in complex dynamical systems.