I am currently working on projects that expand upon the results of my doctoral dissertation. My main areas of focus are that of multifractal analysis and symbolic representations of dynamical systems and their thermodynamic formalism. Many of the dynamical systems that I work with may be represented by countable-state subshifts of finite type. The thermodynamic formalism of which was first developed by Gurewic and Savchenko in the case where the subshift generated by an incidence matrix was assumed to be locally compact. In most cases, the subshifts that I am interested in fail to be locally compact and so the toolbox for understanding these dynamical systems is a bit different. These symbolic dynamical systems are typically associated with some type of underlying iterated function system (IFS), and its corresponding limit set. Many famous ‘fractal’ sets correspond to such limit sets and developing and investigating descriptions of these sets (Hausdorff dimension, connectedness properties, etc.) has become a rapidly evolving area of mathematical research. The study of multifractal analysis, appearing to begin with Rand’s work on cookie-cutter Cantor sets in 1989 and pushed in more recent times by Cawley, Mauldin, and Urbański, amongst others, studies these limit sets by partitioning them based on some dynamically significant parameter hence giving finer descriptions of the limit set than a traditional fractal-dimension analysis could capture.

I have a handful of other projects in preparation as well. I am actively working on problems concerning objects of study in random dynamical systems, random topological dynamics, point-set topology and dimension theory, and nowhere differentiable function theory.