Math B4500: Course Information
Course Title: Dynamical Systems II
Prerequisites: Math A4500.
Catalog Description: Topics will be chosen from the areas of ergodic theory, topological dynamics, differentiable dynamics, complex dynamics and symbolic dynamics.
Semester: Spring 2021.
Meeting time and place: TuTh 6:00 - 7:40pm on Blackboard Collaborate Ultra.
- Name: Prof. Pat Hooper
- Office Hours: Mondays and Tuesdays 1:30-2:30pm on any day classes meet via Zoom. Appointments are also accepted.
The zoom link is listed on blackboard.
- Office: NAC 6/282
- Email: firstname.lastname@example.org
- Clark Robinson. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edition, CRC Press, 1998.
- C. E. Silva, Invitation to Ergodic Theory, American Mathematical Society, 2007.
- Gerald Edgar, Measure, Topology, and Fractal Geometry, 2nd edition, Springer, 2008.
The primary course textbook is the book by Silva. I primarily included Robinson, because it is a broad introduction to the subject and my understanding is that it was used in A4500. The Edgar textbook is feely available from our library through SpringerLink; try the link above. More textbooks and articles may be used in the course, but these will be provided for free to the class (perhaps by the Library).
- Experimentation in the context of dynamical systems (Programming)
- Invariant measures for dynamical systems
- Fractal geometry:
- Fractals and dynamical systems
- Hausdorff measure and dimension
- Integration using measures
- Ergodic theorems
- Interval exchange transformations and Piecewise isometries
- Renormalization (Self-similarity in dynamical systems)
- Notions of mixing (if time permits)
This list of topics is certainly shaped by my own interests and views of what is important in the field of dynamical sytems. I welcome input, and am happy to diverge somewhat from the topics above to address student interests. Dynamics is a broad subject, and while there are broadly applicable approaches and techniques, there is also a diversity of areas of dynamical systems, differing in focus and in goals.
A goal of this class is to introduce the concept of ergodicity and its consequences, which I believe are central to the subject of dynamical systems. In order to do this, we will need some measure theory. The book by Silva offers a basic introduction to measure theory in the context of dynamical systems, which seems ideal. This course is not meant to be a replacement for a Real Analysis course, and I recognize that very few students have seen measure theory before. The goal will be to cover the essential concepts in measure theory carefully, and discuss many of the important and fundamental results, but not rigorously prove most measure results about measure theory so that we have time in the course to devote to dynamical concepts.
General expectations: For each hour spent in the classroom, I expect you to spend at least three hours reading and understanding the book, understanding lecture notes, doing homework, and programming. Practice (doing problems, proofs, and programming) is an important part of learning. Only adequate practice will guarantee that you can complete midterm and final exam problems in a timely manner.
- Homework, Quizzes, and Classwork (40%)
- One Midterm Exam (20%)
- Final Exam (40%)
Exams: Exams will take place on either Zoom or Blackboard Collaborate Ultra. Prof. Hooper reserves the right to spot check anyone's online exam results with a one-on-one oral exam/questioning.
Midterm exam: A midterm will be held on Thu, Mar 25. You will be given the full class to complete the midterm.
Final exam: The final exam will be held at a time determined by the college. As of the writing of this document, this time has not been determined.
Proctoring: This course may employ an online proctoring system for exams, which may require the use of a video camera.
Exam makeups: A midterm missed under well-documented and sufficiently compelling circumstance will result in that grade not counting towards the course grade and the remainder of grades scaled up in value so as to make the total possible grade in the course 100%. (That is in the chart of percentages above, the percentages will be scalled by 100/80.)
If a midterms are missed under well-documented and sufficiently compelling circumstances, an offer of a makeup for the second midterm will be made. The makeup must be taken within one week of the scheduled exam, or a zero will be assigned as the exam grade.
If the final exam is missed under well-documented and sufficiently compelling circumstances, an offer of a makeup for the final exam will be made. The makeup must be taken within one week of the scheduled exam, or a zero will be assigned as the exam grade. The final exam makeup may be an oral exam, even if the regular final exam is not.
Notify me ahead of any exam you expect to miss to be sure your circumstances are sufficiently compelling.