Combinatorial approximations of dynamical systems: a separated graph approach

TL;DR

This paper introduces a combinatorial framework using separated graphs to approximate and analyze dynamical systems on totally disconnected spaces, providing explicit constructions and insights into their structure.

Contribution

It develops a generalized separated graph construction that encodes dynamical systems, allowing explicit disentanglement of space structure and dynamics, extending prior graph-theoretic models.

Findings

Explicit construction of Bratteli-like separated graphs for key examples

Separation of static space structure from dynamics in the graph model

Potential to read minimal systems directly from the separated graph

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.