The algebra of semi-flows: a tale of two topologies

TL;DR

This paper introduces a novel topological framework for dynamical systems, called flow topologies, and develops algebraic invariants through discretization and Morse pre-orders, providing new insights into semi-flows and their invariants.

Contribution

It formulates a bi-topological approach to semi-flows, introduces Morse pre-orders for discretization, and develops algebraic invariants for dynamical systems, especially in the context of parabolic flows.

Findings

Flow topologies effectively describe global dynamics.

Morse pre-orders encode directionality and invariance.

New algebraic invariants for positive braids derived.

Abstract

To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies. We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraization yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows.