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Decompositions of surface flows

Tomoo Yokoyama

TL;DR

This work develops a foundational framework for describing flows on (possibly non-compact) surfaces by constructing complete finite invariants through a decomposition into five invariant open subsets: trivial flow boxes, transverse/periodic annuli, periodic Möbius bands, and locally dense Q-sets. Under mild regularity and tameness assumptions, the authors establish a finite, combinatorial invariant (via labeled graphs) that captures hyperbolic and recurrent dynamics, enabling time-evalutions of fluid-like phenomena to be read as walks on graphs. A key result is that for flows of finite type, the orbit complex serves as a complete invariant, thereby making Markus–Neumann-type descriptions applicable in a broad class including Morse–Smale and generic Hamiltonian flows on compact surfaces; the framework also extends to flows with no-slip boundaries and to non-compact settings. These developments bridge dynamical systems, topology, and computation, offering tools for topological data analysis and the combinatorial study of differential equations in fluid mechanics contexts.

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{ö}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''.

Decompositions of surface flows

TL;DR

This work develops a foundational framework for describing flows on (possibly non-compact) surfaces by constructing complete finite invariants through a decomposition into five invariant open subsets: trivial flow boxes, transverse/periodic annuli, periodic Möbius bands, and locally dense Q-sets. Under mild regularity and tameness assumptions, the authors establish a finite, combinatorial invariant (via labeled graphs) that captures hyperbolic and recurrent dynamics, enabling time-evalutions of fluid-like phenomena to be read as walks on graphs. A key result is that for flows of finite type, the orbit complex serves as a complete invariant, thereby making Markus–Neumann-type descriptions applicable in a broad class including Morse–Smale and generic Hamiltonian flows on compact surfaces; the framework also extends to flows with no-slip boundaries and to non-compact settings. These developments bridge dynamical systems, topology, and computation, offering tools for topological data analysis and the combinatorial study of differential equations in fluid mechanics contexts.

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{ö}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''.

Paper Structure

This paper contains 80 sections, 33 theorems, 12 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Background from a dynamical system's point of view
  4. Background from a recurrent dynamics' point of view
  5. Background from an integrable system's point of view and physical point of view
  6. Background from a computational point of view
  7. Background from a topological dynamics' point of view
  8. Background from a differential equation's point of view
  9. Statements of main results
  10. Detailed descriptions of main results
  11. Preliminaries
  12. Definitions from topology
  13. $T_0$-separation axiom
  14. $T_0$-tification
  15. Dimensions
  16. ...and 65 more sections

Key Result

Theorem 1

Each connected component of $S - \mathop{\mathrm{Bd}}(v)$ for a quasi-regular flow $v$ on a compact surface $S$ is one of the following invariant open subsets exclusively: $(1)$ A trivial flow box in $\mathrm{P}(v)$, whose orbit space is an open interval, $(2)$ An annulus in $\mathrm{P}(v)$, whose o

Figures (14)

  • Figure 1: Multi-saddles, a sink, a $\partial$-sink, a source, a $\partial$-source, and a center.
  • Figure 2: Three fundamental domains: left invariant flow box, middle periodic annulus, right transverse annulus.
  • Figure 3: Examples of multi-saddles
  • Figure 4: A hyperbolic sector with anti-clockwise flow direction and a hyperbolic sector with clockwise flow direction
  • Figure 5: A degenerate singular point with infinitely many connecting separatrices
  • ...and 9 more figures

Theorems & Definitions (174)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 2.1: Lemma 2.1yokoyama2023topological
  • ...and 164 more