# MA4N3 Hyperbolic Dynamics

**Lecturer: **Richard Sharp

**Term(s): **Term 2

**Status for Mathematics students: **List C

**Commitment: **30 one hour lectures

**Assessment: **100% by 3 hour examination

**Formal registration prerequisites:**** **None

**Assumed knowledge:**** **

- Eigenvalues and eigenvectors

MA222 Metric Spaces / MA260 Norms, Metrics and Topologies:

- Metric spaces
- Continuity
- Compactness
- Connectedness
- Cantor sets

- Differentiable functions, diffeomorphisms

- Abstract measures
- Lebesgue measure
- Convergence Theorems
- L^1 and L^2 spaces

**Useful background:**

- Hilbert Spaces
- Dual spaces

- Definition of manifold
- Tangent bundle

- Topological dynamics

**Synergies:**** **

**Leads to:**

**Aims**: This module will have three main strands. First, to understand how hyerbolicity gives rise to complicated dynamical behaviour via the study of accessible examples. Second, to understand the general theory of hyperbolic systems and their properties, such as structural stability, symbolic dynamics and stable manifold theory, and introduce and apply the notion of topological entropy in this setting. Third, to understand more advanced topics and examples, such as equidistribution theory for periodic points, hyperbolic flows, and examples on homogeneous spaces.

**Content**: Hyperbolic Dynamics is a very active area of the field of Dynamical Systems. This course will be an introduction to the subject focusing initially accessible examples which still illustrate the important features of the theory: expanding circle maps, hyperbolic toral automorphisms, Smale horseshoe and Smale solenoid. We will develop key theoretical aspects of the theory of hyperbolic dynamical systems, More advanced topics will be selected from equidistribution theory for periodic points, hyperbolic flows, and examples on homogeneous spaces.

We will cover some of the following topics:

- Expanding maps: expanding circle maps, degree symbolic dynamics, classification, statement of Gromovâ€™s classification theorem, piecewise expanding Markov interval maps, subshifts of finite type.
- Hyperbolic toral automorphisms, Anosov systems, Axion A systems, Smale horseshoe, Smale solenoid, symbolic dynamics, explicit construction of Markov partitions for simple examples.
- Stable manifold theory, shadowing, structural stability.
- Topological entropy, Bowen-Dinaburg definition, invariance under topological conjugacy, calculation for hyperbolic systems.
- Equidistribution of periodic points for hyperbolic toral automorphisms, zeta functions.
- Hyperbolic flows, geodesic flows, suspension flows.
- Flows on matrix groups and homogeneous spaces, connections to number theory.

**Objectives**: By the end of the module, students should be able to:

- Use a variety of analytic and geometric techniques to analyse hyperbolic dynamical systems.
- Understand the role of structural stability in dynamical systems.
- Understand the role of symbolic dynamics in the theory of hyperbolic dynamical systems.
- Understand how topological entropy can be calculated.
- Understand how to study the distribution of orbits of hyperbolic dynamical systems.

**Books: **TBC

**Additional Resources**