General Description
Dynamical systems is the study of iterating a given map.  That is, we take  X  to be some space (for example, an interval, a circle, or something more complicated) and a map T: X ---> X. We then take a point x in X. and repeatedly apply  T, obtaining the sequence of points {x, T(x), T(x)),…}; this is called the orbit of x.
These orbits are generally very complicated.  For example, two points  x  and  y  that start very close to each other may have very different orbits; this is known as sensitive dependence on initial conditions and is the one of the motivations for what has popularly become known as Chaos Theory.
The course starts by describing a number of fundamental examples of dynamical systems, including the doubling map, the continued fraction map and symbolic dynamical systems.  We shall study some of the orbits for these systems; for example, we shall look for periodic orbits (orbits that return to where they started), and dense orbits (orbits that visit every part of  X).
We will also study an important numerical invariant of a dynamical system known as the topological entropy.   Roughly speaking, this measures the complexity of the orbits.
A general dynamical system may be so chaotic that it is impossible to describe every orbit.  Instead, we could attempt to describe what a typical orbit looks like; this is the basis of Ergodic Theory.  To make `typical' precise, we need to use measure theory, and a self-contained introduction to this will be given.  After describing some abstract ergodic theory, we will apply this theory to some of the examples described above. Ergodic theory allows us  to prove several interesting and surprising results in other areas of mathematics.  Here is one example: Consider the sequence 1,2,4,8,16,32,…,2ⁿ,… and construct the sequence of leftmost (or leading) digits:
1,2,4,8,1,3,…    .
How often does the digit 7, say, appear in this sequence?  We will use ergodic theory to prove that about 5.8% of the digits in the above sequence are 7s (the precise answer is log₁₀ 8/7).
The course concludes by discussing Birkhoff's Ergodic Theorem.  This beautiful theorem says that (under appropriate hypotheses!) the proportion of time that a typical orbit spends in some region of X is equal to the measure (area) of that region.  We will apply this theorem to our examples, deriving some interesting and useful corollaries.

Aims
· To obtain an understanding and appreciation of the complexity of the orbit structure of chaotic dynamical systems.
· To work comfortably with invariant measures and ergodic measures.
· To apply these ideas to a number of relevent examples, with particular reference to Birkhoff's Ergodic Theorem.

Prerequisites
251 (ex-UMIST), MT3101 (ex-VUM)

Content
The number in the brackets indicates the approximate number of lectures devoted to that topic.
Introduction [1]
Examples of chaotic dynamical systems [5]
Uniform distribution [3]
An introduction to measure theory [3]
Invariant measures for continuous transformations [5]
Ergodic measures for continuous transformations [5]
Birkhoff's ergodic theorem and applications [2]
Measure-Heoretic entropy and the classification problem [3].

Teaching and learning methods
Two or three lectures each week (alternatively) and fortnightly examples classes.
 

Learning hours

Activity	Hours
Staff/student contact	36
Private study	108
Total hours	144

 

Assessment

Activity	Length	Weighted within unit
End of semester examination	3 hours	100%

Core learning materials
Textbooks
There is not a single recommended textbook.  Some ideas discussed in the first few lectures of the course can be found in R.L. Devaney, An Introduction to Chaotic Dynamical Systems,  Addison-Wesley, 1989.

Good books on ergodic theory include
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1981,
W. Parry, Topics in Ergodic Theory, Cambridge, 1981 (out of print).  Our approach to Ergodic Theory is most closely related to that in Walters' book, although both books contain far more material than is in the course.

A.B. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge, 1995.
This contains all of the material discussed in the course, and much much more.