Combinatorial approximations of dynamical systems: a separated graph approach

Abstract

Separated graphs provide a powerful combinatorial tool for approximating dynamical systems. This paper details the explicit construction of Bratteli-like separated graphs -- a generalization of classical Bratteli diagrams -- that encode the dynamics of a homeomorphism $h$ on a totally disconnected, compact metric space $X$. Unlike standard approaches, the separated graph framework allows us to explicitly disentangle the static structure of the space from the dynamics of the homeomorphism. We provide a step-by-step exposition of this construction applied to four fundamental examples: the two-sided shift, the bit-wise NOT (global flip) map, the classical odometer map and the shift map on the one-point compactification of the integers. Finally, we briefly discuss how minimal (and, more generally, essentially minimal) dynamical systems can be read directly from the separated graph. This approach builds upon recent work by P. Ara and the author, which provides a graph-theoretic model for dynamical systems given by surjective local homeomorphisms defined on totally disconnected compact metric spaces.