MAGIC010: Ergodic Theory

Course details


Spring 2019
Monday, January 21st to Friday, March 29th


Live lecture hours
Recorded lecture hours
Total advised study hours


11:05 - 11:55


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.


  • 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 via a single take-home open-book paper in April with 2 weeks to complete and submit online. You can submit your answers either handwritten or LaTeXed. The deadline for submission will be 12th May. There will be four questions in total and you should answer three of the four questions. If you answer more than three questions then only your three best answers will count. Each question is worth 20 marks. The total number of marks available is 60, and this will then be converted to a percentage. This pass mark is 50 per cent. You should expect to spend no longer than 2hrs on the exam.

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