MAGIC010: Ergodic Theory

Course details

A specialist MAGIC course


Spring 2020
Monday, January 20th to Friday, March 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55 (UK)


A (discrete time) dynamical system consists of a phase space X and a map T : X → X. Dynamical systems concerns studying what happens to points in X under iteration by T. (For notational purposes, write Tn(x) = T °…°T(x), the nth-fold composition of T.) The sequence of points x, T(x), T2(x), …, Tn(x), … is called the orbit of x. Some orbits may be periodic (Tn(x)=x for some n ≥ 1) whereas other orbits may be very complicated and could even be dense in X. Understanding the orbit of a given point x is generally a difficult problem (and is popularly called `chaos').
Ergodic theory takes a more qualitative approach: instead of studying the behaviour of all orbits, we are instead interested in the long-term behaviour of typical orbits. To make `typical' precise one needs to have a measure-theoretic structure on the phase space X; thus ergodic theory can also be viewed as study of dynamical systems in the presence of a measure μ. A basic result of the course is Birkhoff's Ergodic Theorem. Suppose f : X → R is a function. Consider the quantity



which is an average value of f evaluated along the orbit of x. If one regards iteration of T as the passage of time then this quantity can be thought of as a `temporal' average of f along of the orbit of x. Birkhoff's Ergodic Theorem says that for typical points (μ-almost every) x, this temporal average of f is equal to ∫f dμ, a `spatial' average of f.
Ergodic theory has many applications to other areas of mathematics. We will see many connections to problems in metric number theory. For example, we shall use Birkhoff's Ergodic Theorem to study frequencies of digits appearing in number-theoretic expansions (decimals, continued fractions, etc) of real numbers and look at normal numbers.


A good knowledge of metric spaces (to undergraduate level) will be assumed (specifically: continuity, compactness). Familiarity with standard pure mathematics that is taught in UK undergraduate mathematics programmes will be assumed. A knowledge of measure theory will not be assumed and will be introduced in the lectures.

Related courses


  • Lecture 1: Examples of dynamical systems
  • Lecture 2: Uniform distribution mod 1
  • Lecture 3: Invariant measures and measure-preserving transformations
  • Lecture 4: Ergodicity and mixing
  • Lecture 5: Recurrence. Birkhoff's Ergodic Theorem
  • Lecture 6: Topological dynamics
  • Lecture 7: Entropy, information, and the isomorphism problem
  • Lecture 8: Thermodynamic formalism
  • Lecture 9: Applications of thermodynamic formalism: (i) Bowen's formula for Hausdorff dimension, (ii) central limit theorems.
  • Lecture 10: The geodesic flow on compact surfaces of constant negative curvature


  • CW

    Dr Charles Walkden

    University of Manchester


No bibliography has been specified for this course.


The assessment for this course will be released on Monday 20th April 2020 at 00:00 and is due in before Monday 4th May 2020 at 11:00.

The assessment for this course will be a take-home exam. The exam will comprise 4 questions of which your 3 best answers will count. The exam should not take you more than 2 hours. You can either hand-write or LaTeX your answers (whichever is most convenient/fastest for you) and then upload your exam script to the Magic website. The pass mark is 50

Please note that you are not registered for assessment on this course.


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