# Dynamical Systems and Ergodic Theory 34

## 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.

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 (MATH34300). Some of the topics presented have applications in Number Theory (MATH30200) (Gauss map, Weyl’s theorem and equidistributions).

The course will provide a strong foundation from a more pure pespective for more advanced applied courses such as Applied dynamical systems (MATHM0010) and the Engineering unit on Nonlinear Dynamics and Chaos.

Students which took Metric Spaces (MATH20006) 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.

## 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 undertaking postgraduate research in the area.

## Transferable skills

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

## 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 and hyperbolic toral automorphisms;  the Gauss transformation and Continued Fractions;
•  Topol:ogical 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.

## Reading and References

Recommended Reading:

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

Unit code: MATHM6206
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Thomas Jordan

## Pre-requisites

MATH10003 Analysis 1A and MATH10006 Analysis 1B and MATH11007 Calculus 1. This unit may not be taken by students who have taken Dynamical Systems and Ergodic Theory 3.

None

## Methods of teaching

A standard lecture course of lectures and exercises.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

• 90% from a 2 hour 30 minute exam
• 10% assessed coursework

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.