# MAGIC010: Ergodic Theory

## Course details

### Semester

Spring 2018
Monday, January 22nd to Friday, March 16th; Monday, April 23rd to Friday, May 4th

### Hours

Live lecture hours
10
Recorded lecture hours
0
0

Mondays
14:05 - 14:55

## Description

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
 lim n→∞ 1 n n−1∑ j=0 f(Tj(x)),
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.

### Prerequisites

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.

### Syllabus

• 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

## Lecturer

• CW

### Dr Charles Walkden

University
University of Manchester

## Bibliography

No bibliography has been specified for this course.

## Assessment

The assessment for this course will be released on Monday 7th May 2018 and is due in by Friday 25th May 2018 at 23:59.

There will be a take-home exam during the usual Magic examination period. There will be 4 questions of which you have to do 3. You should expect to spend no more than 2 hours on this assessment. Please submit your solutions (handwritten and scanned in, or LaTeXed) via the Magic website.

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