Math 211A (Topics in Analysis)
Course Code: 45025
MWF 1:00 – 1:50
Final Exam: Wednesday, Dec 8, 1:30-3:30pm
Phone: (949) 824-1381
Office Location: RH 510G
Office Hours: Monday 2-3pm or by appointment
Dynamical systems is the study of the long-term behavior of evolving
systems. The modern theory of dynamical systems originated at the end of the
19th century with fundamental question concerning the stability and evolution
of the solar system. Attempts to answer those questions led to the development
of a rich and powerful field with applications to physics, biology,
meteorology, astronomy, economics, and other areas. The mathematical core of
the theory is the study of the global orbit structure of maps and flows with
emphasis on properties invariant under coordinate changes.
This introductory course is aimed at advanced undergraduates, graduate
students, physicists and other non-experts who may want to gain a basic
understanding of the subject.
The following topics will be covered:
examples of topological and smooth dynamics: linear maps, translations on
the torus, gradient flows, expanding maps, symbolic dynamical systems.
Fundamental concepts of dynamical systems: conjugacy, equivalence,
classification, invariants, structural stability.
Local analysis in smooth dynamics: hyperbolic periodic orbits,
Hadamard-Perron theorem, Haryman-Grobman theorem, local structural
stability, normal forms.
dynamics, coding, horseshoes, attractors.
Hyperbolic dynamics: horseshoes, Anosov diffeomorphisms, DA maps, Smale-Williams
solenoid, general hyperbolic sets, Markov partitions, coding, local product
structure, stability, spectral decomposition.
in dynamics. Dynamically defined Cantor sets.
Topological entropy. Calculation of a topological entropy for topological
Markov shifts, hyperbolic automorphisms of the torus, solenoid.
Applications of hyperbolic dynamics to some problems in celestial mechanics
(three body problems) and spectral theory (spectral properties of Fibonacci
B.Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,
G.Stuck, Introduction to Dynamical Systems, Cambridge University
Additional references will be given for a few topics not covered by these
Homework 1 (due Friday, October 8)
Homework 2 (due Friday, October 15)
Homework 3 (due Friday, October 22)
Homework 4 (due Friday, November 5)
Homework 5 (due Friday, November 19)
Homework 6 (due Wednesday, December 1)
Final (take home, due
Friday, December 10)