Current Trends in Dynamical Systems, Fractal Geometry, and Their Applications

Based on AMS session talks, this volume explores the interplay of dynamical systems, ergodic theory, fractal geometry, operator theory, and symbolic dynamics. It presents techniques including relative topological pressure, a dual-operator framework for diffusion, and models in measurable dynamics.

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.