Decompositions of surface flows

Abstract

Flows on surfaces are one of the most fundamental and classical objects in dynamical systems, and are studied from various areas (e.g. integrable systems, differential equations, fluid mechanics). Though hyperbolic flows and recurrent flows on surfaces are classified and characterized using various topological invariants, no complete finite invariants captured both hyperbolicity and recurrence. Moreover, no topological frameworks described even generic time evaluations of gradient flows or incompressible flows (e.g. flows around a circular cylinder placed in uniform flow, solutions of Euler equations and incompressible Navier-Stokes equations). In this paper, to construct a foundation for describing fluid phenomena and capturing hyperbolicity and recurrence, under regularity for the singular point set and tameness of genus and ends, we construct complete finite invariants of flows on (possibly non-compact) surfaces by reconstructing surfaces by gluing five kinds of invariant open subsets, which are trivial flow boxes, transverse/periodic annuli, periodic M{\"o}bius bands, and locally dense Q-sets. Such invariants imply a topological framework that can convert various time evaluations of fluids into walks in graphs without losing topological information, and provide a new tool for analyzing fluid phenomena and differential equations through combinatorics and topological data analysis. Furthermore, such invariants partially revive ``Markus-Neumann theorem''.