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 | Dr. Jordan |
Open unit status | Not open |
Pre-requisites |
MATH10003 Analysis 1A, MATH10006 Analysis 1B, and MATH11007 Calculus 1. |
Co-requisites |
None |
School/department | School of Mathematics |
Faculty | Faculty of Science |
Unit 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.
Unit Description
Dynamical systems is an exciting and very active field in pure and applied mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. A dynamical system can be obtained by iterating a function or letting evolve in time the solution of equation. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it in average.
At the beginning of this lecture course we will give a strong emphasis on presenting many fundamental examples of dynamical systems, such as circle rotations, the baker map on the square and the continued fraction map. Driven by the examples, we will introduce some of the phenomena and main concepts which one is interested in studying.
In the second part of the course, we will formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. We will give full proofs of some of the main theorems.
During the course we will also mention some applications both to other areas of mathematics, such as number theory, and to very concrete problems as data storage and Internet search engines.
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. This unit may not be taken by students who have taken Dynamical Systems and Ergodic Theory 3.
Ergodic Theory has connections with Analysis, Number Theory, Statistical Mechanics and Quantum Chaos.
Students who took MATH20006 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.
The unit will be taught through a combination of
90% Timed, open-book examination 10% Coursework
Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook. Note that the assessed coursework and the final examination will be in part different than the examination for Dynamical Systems and Ergodic Theory 3.
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