Research

The following is a list of my available research works with shortened abstracts:

• Minimal covers with continuity-preserving transfer operators for topological dynamical systems, arxiv:2408.11917. Joint with Kevin Aguyar Brix and Xin Li.

  Abstract: To a non-invertible dynamical system we construct two covers of it by better behaved systems, generalizing the Krieger and Fischer covers of a sub-shift. We show these covers are functorial, have universal properties and study the relationship between properties of the original system and properties of the cover, such as positive expansivity.

• Katsura's self-similar groupoid actions, Putnam's binary factors, and their limit spaces, arxiv:2405.19863 (Invited submission to a special edition of the Journal of the Australian Mathematical Society in honour of Iain Raeburn). Joint with Michael F. Whittaker

  Abstract: We show that the dynamical system associated by Putnam to a pair of graph embeddings is conjugate to the shift on the limit space of a self-similar groupoid action defined by two matrices. We characterize the self-similar properties of these groupoids in terms of properties of the matrices. We prove their limit spaces are bundles of circles and points which fibre over a totally disconnected space and embed into the plane, answering a question posed by Putnam.

• The K-theory of the C*-algebras associated to a rational function, arxiv:2307.13420 (submitted).

  Abstract: We compute the K-theory of the three C*-algebras associated to a rational function, thought of as a dynamical system acting on its Julia set, Fatou set, or the entire Riemann sphere. Our results yield new dynamical invariants for rational functions and a C*-algebraic formulation of the Density of Hyperbolicity Conjecture for quadratic polynomials.

• KK-duality for self-similar groupoid actions on graphs, arxiv:2302.03989 (accepted in Transactions of the American Mathematical Society). Joint with Nathan Brownlowe, Alcides Buss, Daniel Gonçalves, Aidan Sims, and Michael F. Whittaker

  Abstract: We prove that two naturally associated C*-algebras to a regular and contracting self-similar groupoid are Spanier-Whitehead dual (in KK-theory) to each other by showing they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from the self-similar limit space.

• Renormalization procedures for C*-algebras (MSc. Thesis at the University of Victoria), (http://hdl.handle.net/1828/13285). Supervised by Ian F. Putnam.

  Abstract: We introduce renormalization procedures for groupoids and C*-algebras, in analogy to renormalization procedures for families of dynamical systems. We prove a C*-analog to Masur’s unique ergodicity criterion for flat surfaces and apply this criterion to show a variety of C*-algebras have unique trace.