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Dynamical Systems and Ergodic Theory 3 (MATH36206)

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Administrative Information

  1. Unit number and title: MATH36206 Dynamical Systems and Ergodic Theory 3
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2010
  6. Unit Organiser: Corinna Ulcigrai
  7. Lecturer: Dr. Corinna Ulcigrai
  8. Teaching block: 2
  9. Prerequisites: Analysis 1 (MATH11006) and Calculus 1 (MATH 11007).

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.

General Description of the Unit

Dynamical systems is an exciting and very active field in pure and applied mathematics, that involves tools and techniques from many areas such as analyses, 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 Internet search engines.

Relation to Other Units

Ergodic Theory has connections with Analysis, Number theory, Statistical Mechanics and Quantum Chaos. In particular, the course will provide good background 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).

Metric Spaces (MATH 20200) is helpful (students will benefit from some familiarity with metric spaces) but not essential: some basic notions in metric spaces and measure theory will be provided.  

This unit will provide a a pure-mathematical complementary perspective to the Dynamics & Chaos units in applied dynamical systems offered by the Engineering Mathematics Department. 

Teaching Methods

A standard lecture course of 30 lectures and exercises.

Learning Objectives

By the end of the unit the student: 

  • will have  developed an excellent background in the area of dynamical systems,
  • will be familiar with the basic concepts, results, and techniques relevant to the area,
  • will have detailed knowledge of a number of fundamental examples that help clarify and motivate the main concepts in the theory,
  • will understand the proofs of the fundamental theorems in the area,
  • will have mastered the application of dynamical systems techniques for solving a range of standard problems,
  • will have a firm foundation for further study in the area.

Assessment Methods

Standard 2 ½ hour unseen written examination (90%) in May/June together with assessed coursework (10%). The examination will consist of FIVE questions. A candidate's best FOUR answers will be used for assessment.

Award of Credit Points

Credit points for the unit are gained by:

  • either passing the unit (i.e. getting a final assessment mark of 40 or over),
  • or getting a final assesment mark of 30 or over AND handing in satisfactory attempts at the set homework questions.

Transferable Skills

Assimilation of abstract ideas and reasoning in an abstract context. Problem solving and ability to work out model examples.

Texts

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: 

  • B. Hasselblatt and A. Katok, Dynamics: A first course. (Cambdirge University Press, 2003) 
  • M. Brin and G. Stuck, Introduction to Dynamical Systems. (Cambridge University Press, 2002)

Additional references will be given during the course.

Syllabus

  • Basic notions:  dynamical system, orbits, fixed points and fundamental questions;
  • Basic examples of dynamical systems:  circle rotations; the doubling map and expanding maps of the circle; the shift map;  the  Baker’s map;  the CAT map hyperbolic toral automorphisms;  the Gauss transformation and Continued Fractions; 
  • Topological Dynamics:   basic metric spaces notions; minimality;  topological conjugacy;  topological mixing;  topological entropy; topological entropy of toral automorphisms;
  • Symbolic Dynamics:  Shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding; coding of the CAT map;
  • Ergodic Theory:  basic measure theory notions;  invariant measures; Poincare' Recurrence; ergodicity; mixing;  the Birkhoff Ergodic Theorem; Markov measures; Perron-Frobenius theorem, the ergodic theorem for Markov chains and applications to Internet Search. Time permitting: continous time dynamical systems and some mathematical billiards; unique ergodicity;  Weyl’s theorem and applications of recurrence to number theory.