MATH41112 - 2008/2009

 

Specification

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 relevant examples, with particular reference to Birkhoff's Ergodic Theorem.

Brief Description of the unit

Dynamical systems is the study of iterating a given map. That is, we take X to be some mathematical space (for example, an interval, a circle, or perhaps something more complicated) and a map T from X to itself. We then take a point x in X and repeatedly apply T, obtaining the sequence of points {x, T(x), T(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, symbolic dynamical systems, and others. 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 shall also give a geometric description of what the dynamics of these systems look like.

We will also study an important numerical invariant of a dynamical system known as the (measure-theoretic) entropy.

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. We will see that 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 consider 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 (under appropriate hypotheses!) that the proportion of time that a typical orbit spends in some region of X is equal to the measure (area) of the region. We shall then apply this result to our examples, deriving some interesting and useful corollaries.

Learning Outcomes

On successful completion of the course unit students will be able to:

• understand the different kinds of orbits that may arise in the study of dynamical system;
• work with topological entropy and apply it to help decide when two dynamical systems are topologically conjugate;
• understand the basic concepts in ergodic theory, such as measure theory, uniform distribution, invariant measures, ergodicity;
• describe the asymptotic behaviour of ergodic averages via Birkhoff's Ergodic Theorem;
• apply ergodic theory to a number of fundamental examples, rotations on tori, the doubling map, toral automorphisms, the continued fraction map, Bernoulli shifts and Markov shifts.

Future topics requiring this course unit

None

Syllabus

• Introduction
• Examples of chaotic dynamical systems
• Rotations on a circle
• Uniform distribution
• An introduction to measure theory
• Invariant measures for continuous transformations
• Ergodic measures for continuous transformations
• Birkhoff's ergodic theorem and applications
• Information and entropy

Textbooks

There is not a single recommended textbook. Some ideas discussed in the first half 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.

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. The book

• A.B. Katok & B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge, 1995,

also contains material related to the course.

Teaching and learning methods

Two or three lectures per week, fortnightly examples classes.

Assessment

End of semester examination (3 hours) 100%.

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