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.

Figures (12)

• Figure 1: Display of the partitions ${\pazocal P}_0$ through ${\pazocal P}_3$.
• Figure 2: The space $X = {\mathbb Z}^{\ast}$. The integers are mapped onto the circle such that they accumulate at the north pole, representing the point $\infty$.
• Figure 3: The first six partitions ${\pazocal P}_0$ through ${\pazocal P}_5$.
• Figure 6: The first three levels of a separated Bratteli diagram. The blue edges are solid, the red edges are dashed and the green edges are dotted.
• Figure 7: Construction of the edges of $(F,D)$. The blue edges are solid, while the red edges are dashed.

Theorems & Definitions (24)

• Theorem
• Lemma 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.10
• Definition 2.12
• Definition 4.1: Global periodicity for $h$-diagrams