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 T
n(x) = T °…°T(x), the nth-fold composition of T.) The sequence of points x, T(x), T
2(x), …, T
n(x), … is called the
orbit of x. Some orbits may be periodic (T
n(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.
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.