MATH 4290
Dynamical Systems/Ergodic Theory
Fall 2018

This is the homepage for MATH 4290 (Dynamical Systems/Ergodic Theory). This page will be updated throughout the term with important information for our course, including homework assignments, review materials, and more.


  • A topics list/study guide has been posted for your final exam.
  • Solutions to Assignment 7 are posted below.
  • Assignment 8 (your final assignment!) has been posted below.
Course Information

Course meets every TR from 2:00 p.m. - 3:50 p.m. in Knudson Hall, room 100.

Instructor: Ronnie Pavlov
Office: Knudson Hall, room 204
Phone: (303)-871-4001
Office hours: Monday 1-2 and 4-5, Tuesday 4-5, or by appointment.


Text: Introduction to Dynamical Systems by Brin and Stuck.

This book is available at the DU Bookstore.

Course summary

In this course we will study dynamical systems, or deterministic maps on "structured" spaces which preserve "structure." For our purposes, the structured space will always be either a topological space or probability space, and the map will be continuous/measure-preserving respectively.

The main questions we'll consider in this course are long-term behavior of points/sets under repeated iteration of the map, and relationships between different dynamical systems. These questions are actually quite related; some of the most useful invariants for dynamical systems come from long-term behavior of points/sets. We'll cover most basics of topological dynamics, and, as time permits, some important results from measure-theoretic dynamics, such as the celebrated pointwise ergodic theorem. These have some surprising applications; for instance, our topological results allow us to prove that there exists a power of 2 starting with 777, but our measure-theoretic results allow us to find the frequency of such powers of 2!

The prerequisites are real analysis (MATH 3161) and either Real Analysis II (MATH 3162) or Topology (MATH 3110). If you are not sure whether you are a good candidate for the class, please feel free to come talk to me.

Grading scheme

Your term grade will consist of homework assignments, one midterm exam, and one final exam, broken down in the following way:

40% Homework
25% Midterm Exam
35% Final Exam



You will have a midterm exam on Tuesday, October 16th and a final exam on Tuesday, November 20th. Both exams will be in our classroom during classtime (2:00 p.m. - 3:50 p.m.)

Course Policies

  • Students in this course are expected to abide by the University of Denver’s Honor Code and the procedures put forth by the Office of Citizenship and Community Standards. Academic dishonesty - including, but not limited to, plagiarism and cheating - is in violation of the code and will result in a failing grade for the assignment or for the course. As student members of a community committed to academic integrity and honesty, it is your responsibility to become familiar with the DU Honor Code and its procedures: see

  • Students with Disabilities: If you qualify for academic accommodations because of a disability or medical issue, please submit a faculty letter to me from Disability Services Program (DSP) in a timely manner so that your needs may be addressed. DSP determines accommodations based on documented disabilities/medical issues. DSP is located on the 4th floor of Ruffatto Hall, 1999 E Evans Ave, 303-871-2278. Information is also available online at; see the Handbook for Students with Disabilities.

  • Religious Accommodations: University policy grants students excused absences from class or other organized activities for observance of religious holy days, unless the accommodation would create an undue hardship. Faculty are asked to be responsive to requests when students contact them in advance to request such an excused absence. Students are responsible for completing assignments given during their absence, but should be given an opportunity to make up work missed because of religious observance. Once a student has registered for a class, the student is expected to examine the course syllabus for potential conflicts with holy days and to notify the instructor by the end of the first week of classes of any conflicts that may require an absence (including any required additional preparation/travel time). The student is also expected to remind the faculty member in advance of the missed class, and to make arrangements in advance (with the faculty member) to make up any missed work or in-class material within a reasonable amount of time.