Ergodic theory and dimension
Ergodic theory uses invariant measures to characterize certain dynamical systems. Fractional dimensions are used to describe geometrical complexity.
What does it mean to say that a chaotic dynamical system is unpredictable or has some probabilistic features? One answer might be that the time series of such a system is distributed nicely over some set, so that there is an underlying probability distribution for the time evolution of points. A more precise formulation of this idea is that the time averages of some test function (or observation) tend to the integral of the test function over some distribution, so time averages tend to appropriately weighted spatial averages. Understanding the existence and properties of these spatial integrals is at the heart of ergodic theory, and it naturally intersects with measure theory and probability theory.
The invariant objects described by ergodic theory, and those of more general dynamical systems can have infinitely intricate fractal structures. The Hausdorff dimension, and its generalizations or simplifications, can be used to describe geometric objects using an idea of dimension that equals the standard definition for simple objects such as lines or planes, but can be fractional for more complicated objects.
There have been several major advances in ergodic theory over the past few years which make it an exciting time to study this area, and researchers at Manchester work on a wide variety of problems.
Current research areas include:
- Affine measures
- Arithmetic dynamics
- Bernoulli convolutions
- Beta transformations
- Brownian motion
- Piecewise smooth dynamics
- Self-similar sets
- Spectral theory
- Thermodynamic formalism, Gibbs measures and entropy
More information about our research, and some papers, can be found by browsing the webpages of the staff members. Potential PhD students may email staff directly to discuss possible projects.