Ergodic Theory (MAGIC010)
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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
Spring 2017 (Monday, January 23 to Friday, March 31)
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.
No bibliography has been specified for this course.
The assessment for this course will be via a single take-home paper in April with 2 weeks to complete and submit online. The rubric of the exam will be as follows: 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 30 marks. The total number of marks available is 90, and this will then be converted to a percentage. The pass mark is 50exam and you can use the notes provided in the course. There is no time limit, but you should expect to spend no longer than 2hrs on the exam. Submit your answers (either handwritten and then scanned, or LaTeXed) via the Magic website.
MAGIC010 Ergodic Theory exam
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