Zoltán Buczolich:
Introduction to dynamical systems, fractals and ergodic theory
This is an introductory course into Dynamical Systems, Fractals and Ergodic Theory. We start with some examples, physical background and types of dynamical systems. Then Poincaré’s Recurrence Theorem, ergodicity is discussed. We mention without proofs the Lp Ergodic Theorem of von Neumann and Birkhoff’s Ergodic Theorem. We prove that the induced (“derivative”) transformation is measure preserving. We also prove the Kac Lemma. We discuss some definitions concerning topological dynamical systems, the doubling map, ω- and α-limit sets, symbolic dynamical systems, bifurcations.
In the fractals and dynamical systems part of the course we speak about the Mandelbrot set, we introduce Hausdorff-measure and dimension, we mention the Poincaré–Bendixson Theorem and then in higher dimensions some strange attractors, like the Lorenz attractor, we discuss iterated function systems (IFS) and self-similar sets, the logistic family and its bifurcation diagram.