Lectures: TTh 1:30 – 3:05 PM, McHenry Clrm 1279 (the first two weeks remotely)

Instructor: Viktor Ginzburg; office: McHenry 4124, email: ginzburg(at)ucsc.edu

Office Hours: TBA or by appointment

Text: There will be no “official” textbook in this course. Some suggested reading and references:

• Introduction to the Modern Theory of Dynamical Systems by A. Katok and B. Hasselblatt;
• Geometrical Methods in the Theory of Ordinary Differential Equations by V.I. Arnold;
• Lectures on Dynamical Systems by E. Zehnder;
• Measure and Category by J.C. Oxtoby;
• Ergodic Theory by I.P. Cornfeld, S.V. Fomin and Y.G. Sinai;
• Lecture Notes on Ergodic Theory by C. Walkden;
• Dynamical Systems by C. Robinson.

Tentative Syllabus: This course will be a potpourri of dynamical systems, focusing on examples and main concepts and notions rather than technical proofs of general theorems. I plan to discuss or at least briefly touch upon some of the following topics and concepts:

• elements of ergodic theory,
• topological entropy,
• structural stability,
• maps of the circle and the Denjoy example,
• local analysis and local normal forms,
• hyperbolic dynamical systems.

This will not be a comprehensive course in dynamical systems, but rather a non-technical overview of central notions and ideas. Examples are particularly important in dynamics and I will devote a lot of attention to them.

COVID-19 Information: Please make sure to comply with all university guidelines about masking in indoor settings, performing daily symptom and badge checks, testing as required by the campus vaccine policy, self-isolating in the event of exposure, and respecting others’ comfort with distancing. Please do not come to class if your badge is not green. If you are ill or suspect you may have been exposed to someone who is ill, or if you have symptoms that are in any way similar to those of COVID-19, please err on the side of caution and stay home until you are well or have tested negative after an exposure.

Lecture notes (pdf files) The entire set (nearly 100MB). Weekly:

• • Week 1: Basic concepts; Examples: gradient flows, rotations of the circle, translations and linear flows on tori, the Kronecker theorem, geodesic flows
  • Week 2: Examples continued: geodesic flows on surfaces of negative curvature, the shift transformation. Elements of Ergodic Theory: invariant measures, some examples, the Poincare recurrence theorem.
  • Week 3: Elements of Ergodic Theory continued: the Birkhoff Ergodic Theorem; ergodicity and unique ergodicity; Examples: rotations of the circle (equidistribution), translations and linear flows on tori; toral automorphismns.
  • Week 4: Elements of Ergodic Theory continued: mixing; Bernoulli shifts; existence of invariant measures (the Krylov-Bogolubov theorem); the Oxtoby-Ulam theorem.
  • Week 5: Homeomorphisms of the circle: general discussion (equivalence, structurale stability, etc); the rotation number.
  • Week 6: Homeomorphisms of the circle continued: properties of the rotation number; structurally stable diffeomorphisms of the circle; the Denjoy theorem; Digression: continious vs differentiable functions.
  • Weeks 7-8: Homeomorphisms of the circle continued: the Denjoy example; Diophantine vs. Liouville numbers; Herman’s Theorem and small denominators (examples). Local analysis: setting; Lyapunov and asymptotic stability; Lypunov functions.
  • Week 9: Local analysis continued: asymptotic stability via linearization; non-degenerate and hyperbolic fixed points and equilibria; the Hartman-Grobman theorem (without proof); the linearization problem; resonances and the Poincare theorem on formal linearization).
  • Week 10: Introduction to hyperbolic systems: horseshoes; hyperbolic maps and sets; structural stability (Anosov theorem’s on structural stability of hyperbolic toral endomorphisms).