A graph-theoretic characterization of homeomorphisms on the Cantor set

We will explain a graph-theoretic model for dynamical systems given by a homeomorphism on the Cantor set. In order to make the dynamics appear explicitly in the graph, we use two-colored Bratteli separated graphs. This construction can be generalized to dynamical systems given by surjective local homeomorphisms on the Cantor set. We use this construction in order to write the Steinberg algebra associated with the dynamical system as a colimit of the graph algebras associated with the different levels of the corresponding graph. This enables us to compute several invariants of the dynamical system purely in terms of the graph.

This talk will be split into three parts. In the first part I will provide background necessary definitions, and recall the construction of the Leavitt path algebra associated with a separated graph. In the second part I will explain, with an example, how to construct a separated graph out of a dynamical system of our kind. In fact, this construction gives a bijective correspondence between dynamical systems of our kind and a (well-identified) subclass of separated graphs. Finally, in the last part of the talk, I will comment on the relation between the Steinberg algebra, defined purely in terms of the dynamical system, and the Leavitt path algebras of the different levels of the separated graph.

This is joint work with Pere Ara (Universitat Autònoma de Barcelona (UAB), Barcelona).