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Unit information: Dynamical Systems and Ergodic Theory 4 in 2015/16

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Unit name Dynamical Systems and Ergodic Theory 4
Unit code MATHM6206
Credit points 20
Level of study M/7
Teaching block(s) Teaching Block 1 (weeks 1 - 12)
Unit director Professor. Marklof
Open unit status Not open

Analysis 1 (MATH11006) and Calculus 1 (MATH 11007), or equivalent units



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.


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.


  • 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;
  • 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.

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.

Intended learning outcomes

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.

Teaching details

A standard lecture course of 30 lectures and exercises.

Assessment Details

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.

Reading and References

Lecture notes for the course will be provided on a weekly basis and posted on the Course website:

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 might be given during the course.