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Refinements of topological invariants of flows

  • *Corresponding author: Tomoo Yokoyama

    *Corresponding author: Tomoo Yokoyama

This work was partially supported by the JST PRESTO Grant Number JPMJPR16E and JSPS Kakenhi Grant Number 20K03583, 18H01136

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  • We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.

    Mathematics Subject Classification: Primary: 37B35, 37C55; Secondary: 37J46, 37E35, 54B15.

    Citation:

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  • Figure 1.  Reductions of a Hausdorff space into the abstract orbit space

    Figure 2.  Left, the vector field $ X_0 $ on a unit closed ball $ M_0 $; middle, the vector field $ Y_1 $ on a handle $ C $; right, the vector field $ X_1 $ on a solid torus $ M_1 $

    Figure 3.  Left, the vector field $ Y'_1 $; middle, the projection of the vector field $ Y'_1 $ into the $ x $-$ y $ plane; right, the vector field $ X_2 $ on $ M_2 $

    Figure 4.  Left, attaching a copy $ C' $ of the handle $ C $ with the vector field $ Y_1 $; right, attaching a copy $ C' $ of the solid torus $ M_0 \cup C $ with the vector field $ X_1 $

    Figure 5.  Left, attaching two copies of $ C $ to a copy of the ball $ \mathbb{D}^3 $ with the vector field $ -X_0 $; right, attaching a copy of $ C $ to the union of a copy of the ball $ \mathbb{D}^3 $ with the vector field $ -X_0 $ and a copy of $ C $ with the vector field $ -Y_1 $

    Figure 6.  Left, the vector field $ -X_0 $; middle, the vector field $ -Y_1 $; right, the vector field $ X'_1 $ on $ M'_1 $

    Figure 7.  A $ C^\infty $ diffeomorphism on $ \Sigma_2 $

    Figure 8.  A schematic picture of the intersection of the stable manifolds of the saddle $ s_1 $ and the limit cycle $ \gamma $ and the unstable manifolds of the three saddles in $ M_2 $

    Figure 9.  The list of singular points appeared in quasi-regular flows

    Figure 10.  A trivial flow box, an open periodic annulus, and an open transverse annulus

    Figure 11.  Self-connected separatrices and non-self-connected separatrices

    Figure 12.  Commutative digram consisting of canonical projections among an orbit space, a Reeb graph, an abstract weak orbit space, and an extended weak orbit space of a Hamiltonian flow $ v $ with finitely many singular points on a compact surface $ S $

    Figure 13.  Canonical projections and a canonical homeomorphism

    Figure 14.  A neighborhood $ U' $ of the transverse $ T $ and its subsets

    Figure 15.  The picture on the left is a Hamiltonian flow on a closed disk $ \mathbb{D}_1 $. In the diagram on the right, dotted (resp. up, down) directed lines correspond to $ \leq_\pitchfork $ (resp. $ \leq_\alpha $, $ \leq_\omega $)

    Figure 16.  The picture on the left and right are homeomorphisms $ g $ and $ h $ on torus $ \mathbb{T}^2 $ which are time-one map of flows but not topologically equivalent, and whose suspension flows $ v_g $ and $ v_h $ are flows on closed 3-manifolds

    Figure 17.  Two flows generated by Morse-Smale vector fields on a torus are not topologically equivalent but the abstract weak orbit spaces are isomorphic

    Figure 18.  Two flows $ v $ and $ w $ on a disk $ \mathbb{D} $ that are not topologically equivalent and whose multi-saddle connection diagrams are isomorphic as abstract multi-graphs but not isomorphic as plane graphs. In the diagram, dotted (resp. up, down) directed lines correspond to $ \leq_\pitchfork $ (resp. $ \leq_\alpha $, $ \leq_\omega $). In particular, the extended weak orbit spaces are isomorphic and are pre-ordered sets with the pre-orders $ \leq_{v} $ each of which corresponds to the union of the pre-order $ \leq_v $ and the binary relations $ \leq_\alpha $, $ \leq_\omega $ as a direct product

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