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MA424 Dynamical Systems

Lecturer: Richard Sharp

Term(s): Term 1

Status for Mathematics students: List C

Commitment: 30 lectures and weekly assignments

Assessment: 3 hour exam 100%

Formal registration prerequisites: None

Assumed knowledge:

MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

  • Metric and topological spaces
  • Continuous functions
  • Homeomorphisms
  • Compactness
  • The Cantor set

MA259 Multivariable Calculus:

  • Differentiable functions
  • Diffeomorphisms

Useful background:

MA3H5 Manifolds:

  • Definition of manifold
  • Tangent bundle

Synergies: MA427 Ergodic Theory

Content: Dynamical Systems is one of the most active areas of modern mathematics. This course will be a broad introduction to the subject and will attempt to give some of the flavour of this important area.

The course will have two main themes. Firstly, to understand the behaviour of particular classes of transformations. We begin with the study of one dimensional maps: circle homeomorphisms and expanding maps on an interval. These exhibit some of the features of more general maps studied later in the course (e.g., expanding maps, horseshoe maps, toral automorphisms, etc.). A second theme is to understand general features shared by different systems. This leads naturally to their classification, up to conjugacy. An important invariant is entropy, which also serves to quantify the complexity of the system.

Aims: We will cover some of the following topics:

  • circle homeomorphisms and minimal homeomorphisms,
  • expanding maps and Julia sets,
  • horseshoe maps, toral automorphisms and other examples of hyperbolic maps,
  • structural stability, shadowing, closing lemmas, Markov partitions and symbolic dynamics,
  • conjugacy and topological entropy,
  • strange attractors.

Books: R.L. Devaney, An introduction to chaotic dynamical systems, Benjamin.

B.Hasselblat and A.Katok, Dynamics: A first course , CUP, 2003.

S. Sternberg, Dynamical Systems, Dover

Additional Resources