Math 3890-02: Selected Topics for Undergraduates
— Dynamical Systems —
In Spring semester 2019, I am teaching an advanced undergraduate course on Dynamical Systems.
Office hours:
Mondays 2:00–3:30pm Thursdays 2:00–3:30pm and by appointment
Information:
Term: Spring 2019 Time: MWF 12:10–1:00pm Location: 1210 Stevenson Center
Links:
View the Course Syllabus for the course description and course policies. View the Assignments Page for the list of homework assignments. Access our Brightspace page (login required) to find your Grades for the class and other course materials.
Announcements:
- This class will have oral final exams in which students are asked to present proofs of some of the key concepts and results that we have learned during the semester. See the Oral Exam Instructions for details. The exams will be scheduled for the days April 23–25 (Tuesday–Thursday); with each individual examination lasting approximately 30–40 minutes. Please email me to schedule your exam.
- The Eleventh and final Assignment is posted; due in-class on Wednesday April 17.
- The course Brightspace page is up and running. I've posted a copy of my lecture Brightspace. The notes are very unpolished, but you may find them a useful reference or record of what's been discussed in class.
Schedule:
Here is a tentative and ever-evolving schedule of the topics covered in each class. I will aim to have the topics for each class posted at least a class period in advance, and will modify the schedule after the fact to reflect what was actually covered.
Mon Jan 7 — Introduction; Distribute and discuss syllabus; Illustrative examples of dynamics in action (iteration, rotations on a circle, doubling map on a circle, Babylonian algorithm to find square roots)
Wed Jan 9 — More examples of dynamics in action: Newton's method; Fibonacci sequence; first and last diits of powers of 2; Sharkovskii's Theorem; Van der Waerden's Theorem)
Fri Jan 11 — Background: Metric Spaces, examples, open/closed sets; accumulation points; closure; interior.
Mon Jan 14 — Dense, sequences, convergence, completeness, Sequence Lemma.
Wed Jan 16 — Compactness, sequential compactness, Heine–Borel Theorem.
Fri Jan 18 — Student presentations; maps; continuity and its consequences.
Mon Jan 21 — Martin Luther King Day — no class.
Wed Jan 23 — Consqeunes of continuity; Extreme Value Theorem; Lipschitz maps; Contraction Principle.
Fri Jan 25 — Student Presentations of Assignment 2; Contraction Principle.
Mon Jan 28 — Derivative Test and Convexity.
Wed Jan 30 — Stability of pertubations; Eventual and Local Contractions; Newton's Method.
Fri Feb 1 — Student Presentations of Assignment 3.
Mon Feb 4 — Newton's Method; Rotations of the circle; Distribution of orbits in intervals.
Wed Feb 6 — Irrational Rotations: equidistribution of orbits; Birkhoff averagin.
Fri Feb 8 — Student Presentations of Assignment 4.
Mon Feb 11 — Birkhoff averaging for continuous functions; unique ergodicity; distrubtion of digits of powers.
Wed Feb 13 — Lifts, Degree, and Rotation number for circle maps.
Fri Feb 15 — Student Presentations of Assignment 5; proof that rotation number exists.
Mon Feb 18 — Relationship between rotation number and fixed / periodic points.
Wed Feb 20 — Topological conjugacy; Invariance of rotation number; technical lemma.
Fri Feb 22 — Poincare's theorem that minimal homeomorphisms are topologically conjugate to rotations.
Mon Feb 25 — Miterm Exam.
Wed Feb 27 — Denjoy's non-transitive homoemorphisms with irrational rotation number; Expanding maps.
Fri Mar 1 — Expanding maps: growth rate of periodic points; transitivity and mixing.
Mon Mar 11 — Chaotic maps; Sensitivity on initial conditions; Symbolic Dynamics for the doubling map.
Wed Mar 13 — Symbolic Dynamics for expanding maps of the circle; Topological semi-conjugacy.
Fri Mar 15 — Interval Maps; Sarkovskii's Theorem that period 3 implies all periods.
Mon Mar 18 — Sarkovskii's Ordering dictating which periods imply which other periods for interval maps.
Wed Mar 20 — Finish proof of Sarkovskii's Theorem; Converse to Sarkovskii's theorem; Symbolic Dynamics.
Fri Mar 22 — Student Presentations of Assignment 7.
Mon Mar 25 — Basics of Symbolic Dynamics: Shift spaces; Two-sided shift spaces; Shift Maps; Cylinder Sets.
Wed Mar 27 — Smale Horseshoe Map and Topological conjugacy with the two-sided shift.
Fri Mar 29 — Student Presentations of Assignment 8; Subshifts of finte type; Topological Markov Chains.
Mon April 1 — Topological Markov chains: irreducibility, aperiodicity, transitivity, and mixing.
Wed April 3 — Topological Entropy: definition and first examples.
Fri April 5 — Student Presentations of Assignment 9; Conjugacy invariance of entropy.
Mon April 8 — Variations on entropy: separated and spanning sets; Entropy of Topological Markov Chains.
Wed April 10 — Entropy of Topological Markov Chains and spectral radius; Toral automorphisms.
Fri April 12 — Student presentations of Assignment 10; Hyperbolic toral automorphisms.
Mon April 15 — Hyperbolic toral automorphisms: Periodic points, Mixing, Coding, and Markov Partitions.
Wed April 17 — Entropy of hyperbolic toral automorphisms; Shadowing.