Unit name | Dynamical Systems and Ergodic Theory 4 |
---|---|
Unit code | MATHM6206 |
Credit points | 20 |
Level of study | M/7 |
Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
Unit director | Professor. Ulcigrai |
Open unit status | Not open |
Pre-requisites |
Analysis 1 (MATH11006) and Calculus 1 (MATH 11007), or equivalent units |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
Dynamical Systems is an active field in pure and applied mathematics that involves analysis, geometry and number theory. Dynamical systems can be obtained iterating a function or evolving in time the solution of an equation, and often display chaotic long term behaviour. Branches of ergodic theory provide tools to quantify and predict this chaotic behaviour on average. The emphasis in the first part of the unit will be on presenting many fundamental examples of dynamical systems, e.g. rotations, the Baker map, continued fractions. Driven by the examples, it will motivate and introduce key phenomena and concepts. The second part of the unit will formalize the basic definitions and present fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. Proofs the main theorems will be given. Finally the unit will address applications both to other areas of mathematics, such as number theory, and to concrete problems such as data storage and Internet search.
Aims
The course will provide an introduction to subject of dynamical systems, from a pure-mathematical point of view. The first part of the course will be driven by examples so that students will become familiar with various basic models of dynamical systems. We will then develop the mathematical background and the main concepts in topological dynamics, symbolic dynamics and ergodic theory. We will also show applications to other areas of pure mathematics and concrete problems as Internet search.
Syllabus
Relation to Other Units
This is a double-badged version of Dynamical Systems and Ergodic Theory 3, sharing the lectures but with differentiated problems and exam.
Ergodic Theory has connections with Analysis, Number theory, Statistical Mechanics and Quantum Chaos. In particular, the course will provide good mathematical foundations in dynamics for students interested in Statistical Mechanics (MATH 34300). Some of the topics presented have applications in Number Theory (MATH 30200) (Gauss map, Weyl’s theorem and equidistributions).
Students which took MATH 20200 Analysis 2 (Metric Spaces) will benefit from some familiarity with metric spaces, but students who did not will be provided with basic notions in metric spaces and measure theory.
This unit will provide a a pure-mathematical complementary perspective to the Dynamics & Chaos unit in applied dynamical systems offered by the Engineering Mathematics Department.
By the end of the unit the student :
Transferable Skills:
Assimilation of abstract ideas and reasoning in an abstract context. Problem solving and ability to work out model examples.
A standard lecture course of 30 lectures and exercises.
Standard 2 ½ hour unseen written examination (90%) in April together with assessed coursework (10%). The examination consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment.
Note that the assessed coursework and the final examination will be in part different than the examination for Dynamical Systems and Ergodic Theory 3.
Lecture notes for the course will be provided on a weekly basis and posted on the Course website:
http://www.maths.bristol.ac.uk/~maxcu/DynSysErgTh.html
In addition, useful textbook are:
Additional references might be given during the course.