A Complete Invariant for Flow Equivalence

TL;DR

This paper establishes a comprehensive invariant for classifying flow equivalence of minimal one-dimensional flow spaces, linking dynamical systems, symbolic systems, and algebraic invariants.

Contribution

It introduces a complete invariant based on positive trope classes of inverse sequences of free groups, extending prior classification results.

Findings

Provides a one-to-one correspondence between flow classes and symbolic system invariants.

Realizes every symbolic system as a flow space, ensuring completeness.

Extends classification frameworks for primitive substitution tilings and Z-Cantor systems.

Abstract

Minimal flow spaces of dimension 1 are among the most fundamental limit sets in dynamical systems. These invariant sets occur as the typical minimal sets in surface flows, the minimal sets of suspensions of subshifts (for example, in Lorenz template models of the Lorenz attractor) and the hulls of repetitive tilings of dimension one. Here we establish a complete invariant for the flow equivalence of such objects. The invariant takes values in a category of `positive trope' classes of inverse sequences of free groups and positive maps, or alternatively within a certain category of symbolic systems. Moreover, every such symbolic system is realised by a flow space, and we thus have a one to one correspondence between flow equivalence classes of minimal flow spaces and positive trope classes of such systems. At the same time, this provides a complete invariant both for flow equivalence of minimal Z-Cantor dynamical systems and for germinal equivalence of minimal Z-Cantor systems. Our work thus greatly extends that of Barge and Diamond on their complete invariant of primitive substitution tilings, and provides counterpoint to the work of Giordano, Putnam and Skau on orbit equivalence of Z-Cantor systems.