Featured presentations from notable and upcoming researchers in the field of Dynamical Systems
Computability in dynamical systems is a relatively young field that has attracted significant attention in recent years. One of its central questions is whether dynamically relevant objects can be algorithmically represented by a Turing machine. While this question has been extensively studied in symbolic dynamics, where computability results are known for various thermodynamical quantities such as entropy, pressure, equilibrium states and zero-temperature measures, a corresponding general theory for broader classes of topological and smooth dynamical systems is lacking. In this talk, we present an approach to bridging this gap by introducing the concept of computable Markov partitions. This framework allows us to establish far-reaching computability results for several classes of topological and smooth dynamical systems. The results presented in this talk are joint work with Michael Burr and Tamara Kucherenko.
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The Mandelbrot set encodes the dynamical dependence of quadratic polynomials on a parameter. The MLC Conjecture (asserting that the Mandelbrot set is locally connected) is a rigidity property that yields a satisfactory topological description of the Mandelbrot set. In this talk, we describe the historical motivation for the conjecture, explain how it became a central topic in Renormalization Theory (analyzing first-return maps to small neighborhoods of special points), and outline some of the ideas behind the most recent advances toward MLC and related questions.
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