# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021191
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## Refinements of topological invariants of flows

 Applied Mathematics and Physics Division, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan

*Corresponding author: Tomoo Yokoyama

Received  July 2021 Revised  October 2021 Early access December 2021

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

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.

Citation: Tomoo Yokoyama. Refinements of topological invariants of flows. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021191
##### References:
 [1] A. Abbondandolo and P. Majer, Stable foliations and CW-structure induced by a Morse-Smale gradient-like flow, To appear in J. Topol. Anal., arXiv preprint, arXiv: 2003.07134, (2020). Google Scholar [2] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789.  doi: 10.1137/080734935.  Google Scholar [3] S. K. Aranson, G. R. Belitskiĭ and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, (1996).  doi: 10.1090/mmono/153.  Google Scholar [4] M. Audin and M. Damian, Morse Theory and Floer Homology, Springer, (2014).  doi: 10.1007/978-1-4471-5496-9.  Google Scholar [5] G. D. Birkhoff, Dynamical Systems, With an Addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. Ⅸ, American Mathematical Society, Providence, R.I., 1966.  Google Scholar [6] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, CRC Press, 2004. doi: 10.1201/9780203643426.  Google Scholar [7] C. Bonatti, H. Hattab and E. Salhi, Quasi-orbit spaces associated to $T_0$-spaces, Fund. Math., 211 (2011), 267-291.  doi: 10.4064/fm211-3-4.  Google Scholar [8] C. Bonatti, H. Hattab, E. Salhi and G. Vago, Hasse diagrams and orbit class spaces, Topology Appl., 158 (2011), 729-740.  doi: 10.1016/j.topol.2010.12.010.  Google Scholar [9] J. G. E. Buendía, H. Hattab and V. J. López, On the Markus–Neumann theorem, Journal of Differential Equations, 265 (2018), 6036-6047.  doi: 10.1016/j.jde.2018.07.021.  Google Scholar [10] M. Cobo, C. Gutierrez and J. Llibre, Flows without wandering points on compact connected surfaces, Transactions of the American Mathematical Society, 362 (2010), 4569-4580.  doi: 10.1090/S0002-9947-10-05113-5.  Google Scholar [11] C. Conley, The gradient structure of a flow: Ⅰ, Ergodic Theory and Dynamical Systems, 8 (1988), 11-26.  doi: 10.1017/S0143385700009305.  Google Scholar [12] C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Soc., (1978).   Google Scholar [13] J. M. Franks, Morse-Smale flows and homotopy theory, Topology, 18 (1979), 199-215.  doi: 10.1016/0040-9383(79)90003-X.  Google Scholar [14] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory and Dynamical Systems, 6 (1986), 17-44.  doi: 10.1017/S0143385700003278.  Google Scholar [15] S. Harker, K. Mischaikow, M. Mrozek and V. Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math., 14 (2014), 151-184.  doi: 10.1007/s10208-013-9145-0.  Google Scholar [16] G. Kalmbach, On some results in Morse theory, Canadian Journal of Mathematics, 27 (1975), 88-105.  doi: 10.4153/CJM-1975-011-0.  Google Scholar [17] H. Kantz, Schreiber, Thomas, Nonlinear Time Series Analysis, Cambridge university press, 7 1997.  Google Scholar [18] R. Labarca and M. J. Pacifico, Stability of Morse-Smale vector fields on manifolds with boundary, Topology, 29 (1990), 57-81.  doi: 10.1016/0040-9383(90)90025-F.  Google Scholar [19] F. Laudenbach, On the Thom-Smale complex, Astérisque, 205 (1992), 219-233. Google Scholar [20] T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, American Mathematical Soc., (2005).  doi: 10.1090/surv/119.  Google Scholar [21] L. Markus, Global structure of ordinary differential equations in the plane, Transactions of the American Mathematical Society, 76 (1954), 127-148.  doi: 10.1090/S0002-9947-1954-0060657-0.  Google Scholar [22] M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J., 33 (1966), 465-474.   Google Scholar [23] K. R. Meyer, Energy functions for Morse-Smale systems, American Journal of Mathematics, 90 (1968), 1031-1040.  doi: 10.2307/2373287.  Google Scholar [24] J. Milnor, Morse Theory, Based on Lecture Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963.  Google Scholar [25] K. Mischaikow, M. Mrozek and F. Weilandt, Discretization strategies for computing Conley indices and Morse decompositions of flows, J. Comput. Dyn., 3 (2016), 1-16.  doi: 10.3934/jcd.2016001.  Google Scholar [26] J. Montgomery, Cohomology of isolated invariant sets under perturbation, Journal of Differential Equations, 13 (1973), 257-299.  doi: 10.1016/0022-0396(73)90018-1.  Google Scholar [27] M. Mrozek, R. Srzednicki and F. Weilandt, A topological approach to the algorithmic computation of the Conley index for Poincaré maps, SIAM J. Appl. Dyn. Syst., 14 (2015), 1348-1386.  doi: 10.1137/15M100794X.  Google Scholar [28] D. Neumann and T. O'Brien, Global structure of continuous flows on 2-manifolds, Journal of Differential Equations, 22 (1976), 89-110.  doi: 10.1016/0022-0396(76)90006-1.  Google Scholar [29] D. A. Neumann, Classification of continuous flows on 2-manifolds, Proceedings of the American Mathematical Society, (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.  Google Scholar [30] I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds: An overview, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1705, 1999. doi: 10.1007/BFb0093599.  Google Scholar [31] E. Ott, T. Sauer and J. A. Yorke, Coping with chaos. analysis of chaotic data and the exploitation of chaotic systems, Wiley Series in Nonlinear Science, (1994), 1-62.   Google Scholar [32] J. Palis, Structural stability theorems, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 223-231.   Google Scholar [33] J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-404.  doi: 10.1016/0040-9383(69)90024-X.  Google Scholar [34] J. Palisb and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., vol. ⅹⅳ, Berkeley, Calif., 1968), 223–231.  Google Scholar [35] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (ⅰ), Journal de Mathématiques Pures et Appliquées, 7 (1881), 375-422.   Google Scholar [36] L. Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, Journal of Topology and Analysis, 2 (2010), 469-526.  doi: 10.1142/S1793525310000409.  Google Scholar [37] L. Qin, An application of topological equivalence to Morse theory, J. Fixed Point Theory Appl., 23 (2021), Paper No. 10, 38 pp. arXiv preprint, arXiv: 1102.2838, (2011). doi: 10.1007/s11784-020-00843-z.  Google Scholar [38] S. Smale, On gradient dynamical systems, Annals of Mathematics, 74 (1961), 199-206.  doi: 10.2307/1970311.  Google Scholar [39] F. Takens, Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick, 1980, Springer, 366–381.  Google Scholar [40] R. Thom, Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris, 228 (1949), 973-975.   Google Scholar [41] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin (new series) of the American Mathematical Society, 19 (1988), 417-431.  doi: 10.1090/S0273-0979-1988-15685-6.  Google Scholar [42] H. Whitney, Differentiable manifolds, Annals of Mathematics, 37 (1936), 645-680.  doi: 10.2307/1968482.  Google Scholar [43] T. Yokoyama, Decompositions of surface flows, arXiv preprint, arXiv: 1703.05501, (2017). Google Scholar [44] T. Yokoyama, Properness of foliations, Topology and its Applications, 254 (2019), 171-175.  doi: 10.1016/j.topol.2018.12.008.  Google Scholar [45] T. Yokoyama and T. Sakajo, Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20120558.  doi: 10.1098/rspa.2012.0558.  Google Scholar

show all references

##### References:
 [1] A. Abbondandolo and P. Majer, Stable foliations and CW-structure induced by a Morse-Smale gradient-like flow, To appear in J. Topol. Anal., arXiv preprint, arXiv: 2003.07134, (2020). Google Scholar [2] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., 8 (2009), 757-789.  doi: 10.1137/080734935.  Google Scholar [3] S. K. Aranson, G. R. Belitskiĭ and E. V. Zhuzhoma, Introduction to the Qualitative Theory of Dynamical Systems on Surfaces, American Mathematical Society, (1996).  doi: 10.1090/mmono/153.  Google Scholar [4] M. Audin and M. Damian, Morse Theory and Floer Homology, Springer, (2014).  doi: 10.1007/978-1-4471-5496-9.  Google Scholar [5] G. D. Birkhoff, Dynamical Systems, With an Addendum by Jurgen Moser. American Mathematical Society Colloquium Publications, Vol. Ⅸ, American Mathematical Society, Providence, R.I., 1966.  Google Scholar [6] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification, CRC Press, 2004. doi: 10.1201/9780203643426.  Google Scholar [7] C. Bonatti, H. Hattab and E. Salhi, Quasi-orbit spaces associated to $T_0$-spaces, Fund. Math., 211 (2011), 267-291.  doi: 10.4064/fm211-3-4.  Google Scholar [8] C. Bonatti, H. Hattab, E. Salhi and G. Vago, Hasse diagrams and orbit class spaces, Topology Appl., 158 (2011), 729-740.  doi: 10.1016/j.topol.2010.12.010.  Google Scholar [9] J. G. E. Buendía, H. Hattab and V. J. López, On the Markus–Neumann theorem, Journal of Differential Equations, 265 (2018), 6036-6047.  doi: 10.1016/j.jde.2018.07.021.  Google Scholar [10] M. Cobo, C. Gutierrez and J. Llibre, Flows without wandering points on compact connected surfaces, Transactions of the American Mathematical Society, 362 (2010), 4569-4580.  doi: 10.1090/S0002-9947-10-05113-5.  Google Scholar [11] C. Conley, The gradient structure of a flow: Ⅰ, Ergodic Theory and Dynamical Systems, 8 (1988), 11-26.  doi: 10.1017/S0143385700009305.  Google Scholar [12] C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Soc., (1978).   Google Scholar [13] J. M. Franks, Morse-Smale flows and homotopy theory, Topology, 18 (1979), 199-215.  doi: 10.1016/0040-9383(79)90003-X.  Google Scholar [14] C. Gutiérrez, Smoothing continuous flows on two-manifolds and recurrences, Ergodic Theory and Dynamical Systems, 6 (1986), 17-44.  doi: 10.1017/S0143385700003278.  Google Scholar [15] S. Harker, K. Mischaikow, M. Mrozek and V. Nanda, Discrete Morse theoretic algorithms for computing homology of complexes and maps, Found. Comput. Math., 14 (2014), 151-184.  doi: 10.1007/s10208-013-9145-0.  Google Scholar [16] G. Kalmbach, On some results in Morse theory, Canadian Journal of Mathematics, 27 (1975), 88-105.  doi: 10.4153/CJM-1975-011-0.  Google Scholar [17] H. Kantz, Schreiber, Thomas, Nonlinear Time Series Analysis, Cambridge university press, 7 1997.  Google Scholar [18] R. Labarca and M. J. Pacifico, Stability of Morse-Smale vector fields on manifolds with boundary, Topology, 29 (1990), 57-81.  doi: 10.1016/0040-9383(90)90025-F.  Google Scholar [19] F. Laudenbach, On the Thom-Smale complex, Astérisque, 205 (1992), 219-233. Google Scholar [20] T. Ma and S. Wang, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, American Mathematical Soc., (2005).  doi: 10.1090/surv/119.  Google Scholar [21] L. Markus, Global structure of ordinary differential equations in the plane, Transactions of the American Mathematical Society, 76 (1954), 127-148.  doi: 10.1090/S0002-9947-1954-0060657-0.  Google Scholar [22] M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J., 33 (1966), 465-474.   Google Scholar [23] K. R. Meyer, Energy functions for Morse-Smale systems, American Journal of Mathematics, 90 (1968), 1031-1040.  doi: 10.2307/2373287.  Google Scholar [24] J. Milnor, Morse Theory, Based on Lecture Notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963.  Google Scholar [25] K. Mischaikow, M. Mrozek and F. Weilandt, Discretization strategies for computing Conley indices and Morse decompositions of flows, J. Comput. Dyn., 3 (2016), 1-16.  doi: 10.3934/jcd.2016001.  Google Scholar [26] J. Montgomery, Cohomology of isolated invariant sets under perturbation, Journal of Differential Equations, 13 (1973), 257-299.  doi: 10.1016/0022-0396(73)90018-1.  Google Scholar [27] M. Mrozek, R. Srzednicki and F. Weilandt, A topological approach to the algorithmic computation of the Conley index for Poincaré maps, SIAM J. Appl. Dyn. Syst., 14 (2015), 1348-1386.  doi: 10.1137/15M100794X.  Google Scholar [28] D. Neumann and T. O'Brien, Global structure of continuous flows on 2-manifolds, Journal of Differential Equations, 22 (1976), 89-110.  doi: 10.1016/0022-0396(76)90006-1.  Google Scholar [29] D. A. Neumann, Classification of continuous flows on 2-manifolds, Proceedings of the American Mathematical Society, (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.  Google Scholar [30] I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds: An overview, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1705, 1999. doi: 10.1007/BFb0093599.  Google Scholar [31] E. Ott, T. Sauer and J. A. Yorke, Coping with chaos. analysis of chaotic data and the exploitation of chaotic systems, Wiley Series in Nonlinear Science, (1994), 1-62.   Google Scholar [32] J. Palis, Structural stability theorems, Global Analysis, Proc. Sympos. Pure Math., 14 (1970), 223-231.   Google Scholar [33] J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-404.  doi: 10.1016/0040-9383(69)90024-X.  Google Scholar [34] J. Palisb and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., vol. ⅹⅳ, Berkeley, Calif., 1968), 223–231.  Google Scholar [35] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (ⅰ), Journal de Mathématiques Pures et Appliquées, 7 (1881), 375-422.   Google Scholar [36] L. Qin, On moduli spaces and CW structures arising from Morse theory on Hilbert manifolds, Journal of Topology and Analysis, 2 (2010), 469-526.  doi: 10.1142/S1793525310000409.  Google Scholar [37] L. Qin, An application of topological equivalence to Morse theory, J. Fixed Point Theory Appl., 23 (2021), Paper No. 10, 38 pp. arXiv preprint, arXiv: 1102.2838, (2011). doi: 10.1007/s11784-020-00843-z.  Google Scholar [38] S. Smale, On gradient dynamical systems, Annals of Mathematics, 74 (1961), 199-206.  doi: 10.2307/1970311.  Google Scholar [39] F. Takens, Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick, 1980, Springer, 366–381.  Google Scholar [40] R. Thom, Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris, 228 (1949), 973-975.   Google Scholar [41] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bulletin (new series) of the American Mathematical Society, 19 (1988), 417-431.  doi: 10.1090/S0273-0979-1988-15685-6.  Google Scholar [42] H. Whitney, Differentiable manifolds, Annals of Mathematics, 37 (1936), 645-680.  doi: 10.2307/1968482.  Google Scholar [43] T. Yokoyama, Decompositions of surface flows, arXiv preprint, arXiv: 1703.05501, (2017). Google Scholar [44] T. Yokoyama, Properness of foliations, Topology and its Applications, 254 (2019), 171-175.  doi: 10.1016/j.topol.2018.12.008.  Google Scholar [45] T. Yokoyama and T. Sakajo, Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20120558.  doi: 10.1098/rspa.2012.0558.  Google Scholar
Reductions of a Hausdorff space into the abstract orbit space
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 $M_1$
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 $M_2$
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 $X_1$
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 $-Y_1$
Left, the vector field $-X_0$; middle, the vector field $-Y_1$; right, the vector field $X'_1$ on $M'_1$
A $C^\infty$ diffeomorphism on $\Sigma_2$
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 $M_2$
The list of singular points appeared in quasi-regular flows
A trivial flow box, an open periodic annulus, and an open transverse annulus
Self-connected separatrices and non-self-connected separatrices
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 $S$
Canonical projections and a canonical homeomorphism
A neighborhood $U'$ of the transverse $T$ and its subsets
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. $\leq_\alpha$, $\leq_\omega$)
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
Two flows generated by Morse-Smale vector fields on a torus are not topologically equivalent but the abstract weak orbit spaces are isomorphic
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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