Description
Objectives
Give a broad introduction to fundamental topics of dynamical systems, in addition to those already discussed in Ordinary Differential Equations.
Syllabus
Basic notions: dynamical system, rotations of the circle, translations of the interval, autonomous differential equations and flows, expanding transformations, toral automorphisms. Topological dynamics: limit sets, topological transitivity and topological mixing, topological entropy, expanding transformations, hyperbolic toral automorphisms. Dimensional dynamics: diffeomorphisms of the circle, rotation number, topological conjugacy, Denjoy's theorem, interval maps, Sharkovsky's order. Hyperbolic dynamics: hyperbolic sets, invariant cones, stable and unstable invariant manifolds, Smale horseshoes, product structure, geodesic flows. Symbolic dynamics: shifts and subshifts, topological Markov chains, product topology, topological transitivity, topological entropy. Ergodic theory: invariant measures, examples with the Lebesgue measure, Gauss map, Poincaré's recurrence theorem and applications, metric entropy.
Prerequisites
Ordinary Differential Equations.
Cross Competence Component
The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.
Ethical Principles
All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.