# 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.

**Section:** ST.

**Meeting time and place:** TuTh 6:00 - 7:40pm on Blackboard Collaborate Ultra.

**Instructor Information:**

**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:**whooper@ccny.cuny.edu

**Course Textbooks:**

- 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).

**Topics Covered:**A

*tentative*list of topics covered is given below. I reserve the right to alter the topics in the course. (This could be necessary because of student interests, background, available time, etc.) This is not meant to be an exhaustive list of topics covered.

- Measures
- Experimentation in the context of dynamical systems (Programming)
- Invariant measures for dynamical systems
- Fractal geometry:
- Fractals and dynamical systems
- Hausdorff measure and dimension

- Recurrence
- Ergodicity
- 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.

**Expectations of written work:**Mathematical proofs and calculations will be graded partially on presentation. In order to receive full credit, a student who reads your answer should be able to easily understand how you solved the problem. Written work is expected to be legible and arguments are expected to be well articulated.

**Grades:**Grades will be computed from the following weighted average:

- 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.

**Homework assignments:**Homework will be assigned approximately once a week and will have a due date. Homework assignments will be made available on the course website at least one week before the assignment is due. I encourage you to work in groups on the homework problems, especially if this best suits your learning style. Nonetheless, you should be confident that you understand how to do each problem, and should be able to solve similar problems independently. Failure to ensure that you can solve problems independently will surely have a negative effect on exam grades.

**Late homework:**Late homework will not be accepted for any reason.

**Quizzes:**There will be occasional quizzes to test that you are keeping up with the course.

**Dropped grades:**The lowest 20% of assignments in the “Homework, Quizzes, and Classwork” category will be dropped (rounding up if necessary).

**Course website:**Course information, homework assignments, and documents can be found on the website:

**Blackboard:**I use blackboard to keep track of your grades. You can view your grades there. To access blackboard visit: I will likely use blackboard collaborate for class meetings. Solutions to problem sets will be posted to blackboard.

**Zoom:**I have been using Zoom for office hours. A link will be sent by email.

**Video lectures:**Students who participate in this class with their camera on or use a profile image are agreeing to have their video or image recorded solely for the purpose of creating a record for students enrolled in the class to refer to, including those enrolled students who are unable to attend live. If you are unwilling to consent to have your profile or video image recorded, be sure to keep your camera off and do not use a profile image. Likewise, students who un-mute during class and participate orally are agreeing to have their voices recorded. If you are not willing to consent to have your voice recorded during class, you will need to keep your mute button activated and communicate exclusively using the “chat” feature, which allows students to type questions and comments live.

**Academic integrity:**You are expected to adhere to the CUNY Policy on Academic Integrity. This policy is posted at In particular, it is expected that you not plagarize. This has been especially problematic on homework. Your homework must not copy from another source, and you must cite any sources used when preparing your solutions. Sources can include textbooks, webpages, discussions with other people, and other student work. Citations should be as specific as possible. All submitted work must be written in your own words.