Fractal Geometry and Dimension Theory
Roughly speaking, ‘fractals’ are sets which exhibit interesting structures at arbitrarily small scales. Such objects appear naturally across science and studying them in a rigorous mathematical framework is of great interest. Falconer’s textbook: Fractal Geometry: Mathematical Foundations and Applications is a standard reference in the field and is used for teaching and research across the world. Many fractals, including self-similar and self-affine sets, are often modelled as attractors of iterated function systems. We are interested in various properties of these sets, including the relationship between their geometry and their dimension theory. Falconer pioneered the study of the Hausdorff dimension of self-affine sets in the 1980s, but this topic is still very much in vogue today. Indeed, Falconer and Kempton have recently been involved in linking the problem to dynamically defined measures on an associated projective space. Fraser has investigated Assouad dimension in a variety of contexts, including showing that there is no direct analogue of Marstrand’s Projection Theorem for Assouad dimension.
Dynamical Systems and Ergodic Theory
Dynamical systems is the study of systems which evolve in time (discrete or continuous) with a view to understanding the long-term behaviour. However, these systems often exhibit `chaotic’ signatures, rendering a complete understanding impossible. Ergodic theory can be applied to these problems, taking a probabilistic viewpoint to investigate the average statistical behaviour of the system. The fundamental objects here are measures: which ones are important, and, what happens for typical points for these measures under the dynamics. The speed at which the system begins to look `completely random’ is a key signature of the chaotic behaviour. Thermodynamic formalism provides a toolbox for investigating interesting measures, speeds of mixing, and probabilistic laws. Todd uses these ideas to understand systems which mix very slowly (the `least chaotic’ systems), or do not quite mix at all (transient systems). These are particularly interesting because the intermittency (predictable phases interspersed with chaotic bursts) these systems exhibit is seen in numerous real-world applications. Todd’s main interest is in understanding the statistical behaviour of elementary models, in order to investigate what should be the signatures of the above-mentioned more sophisticated models.
The measure-theoretic multifractal formalism introduced in the 1990s by Olsen is now standard in rigourous multifractal analysis and has been used in many questions involving analysis of measures. Topics we study from a multifractal viewpoint include: self-affine measures, statistically self-similar measures, divergence points, points of non-differentiability of functions, and inhomogeneous measures. More recently, Olsen has used zeta functions to analyse the multifractal properties of self-similar and self-conformal measures, bringing together ideas from dynamical systems, complex analysis and geometric measure theory.
Stochastic processes, including Gaussian and stable processes, are used increasingly to model highly irregular phenomena. Our recent work on fractal stochastic processes has concerned the localisibility of such processes, that is, their local form when scaled about particular points. In collaboration with French mathematicians Falconer has constructed ‘multistable processes’, where the stability index varies with time, and `self-regulating processes’ where the stability index depends on the value of the process at any instant. Related work has involved studying the Fourier transform of measures supported on graphs of random functions, such as Brownian and fractional Brownian motion. Fraser and his collaborators proved that such graphs are almost surely not Salem, i.e. the Hausdorff dimension is not witnessed by Fourier decay.
Group Actions, Dynamics, and Group Theory
Automorphisms of one- and two-sided shift spaces, and the R. Thompson family of groups are central examples of groups of homeomorphisms of spaces, which have a long pedigree of relevance to many disparate fields of mathematics research. Bleak and his collaborators have analysed these groups through studying properties of their actions on relevant Cantor spaces. Techniques used in this study come from many areas including: algebra, theoretical computer science, combinatorics, symbolic dynamics, and analysis.