Introduction to Dynamical Systems

This is Part I in a planned two-part year-long course "Introdcution to Dynamical Systems". The main objective of the course is to introduce several key branches of modern dynamical systems to the IST community. Part I, which will be taught in the fall semester of 2022, will roughly follow Chapters 1-6 of "Introduction to Dynamical Systems" by M. Brin, G. Stuck (Cambridge Univ. Press, 2002). We will start with exploring explicit examples of dynamical systems. We will then gradually revisit these examples as we progress in building up theory of symbolic dynamics, hyperbolic dynamical systems, Anosov systems, and ergodic theory. We plan to cover: -- various exlicit examples of dynamical systems (circle maps, real maps of the interval, complex one-dimensional maps, toral automorphisms, horshoes and solenoids, etc); -- elements of symbolic dynamics (subshifts, entropy, application to coding); -- elements of hyperbolic dynamical systems, mainly in R^n (stable/unstable manifolds, stability, density of hyperbolicity and Axiom A, Markov partitions); -- elements of ergodic theory (classical ergodic theorems, application to number theory, ergodicity of Anosov flows). We will outline several applications of the circle of ideas that will be discussed during the semester to other areas of mathematics (primarily, to geometry, number theory, algebra, ordinary differential equations, iterative algorithms).