# American Institute of Mathematical Sciences

September 2018, 38(9): 4305-4327. doi: 10.3934/dcds.2018188

## Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs

 HSE; Bolshaya Pecherskaya 25/12, Nizhniy Novgorod, 603155, Russia

* Corresponding author: Olga Pochinka

Received  May 2017 Revised  April 2018 Published  June 2018

Fund Project: Authors are grateful to participants of the seminar "Topological Methods in Dynamics" for fruitful discussions. The classification results (Sections 1–6 without Subsections 5.2, 5.3) were obtained with the support of the Russian Science Foundation (project 17-11-01041). The realisation results (Subsection 5.2, Section 7) were obtained as an output of the research project "Topology and Chaos in Dynamics of Systems, Foliations and Deformation of Lie Algebras (2018)" implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE). The algorithmic results (Subsection 5.3, Section 8) were obtained with the support of Russian Foundation for Basic Research 16-31-60008-mol-a-dk and with LATNA laboratory, National Research University Higher School of Economics

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to $Ω$-stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.

Citation: Vladislav Kruglov, Dmitry Malyshev, Olga Pochinka. Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4305-4327. doi: 10.3934/dcds.2018188
##### References:
 [1] V. E. Alekseev and V. A. Talanov, Graphs and Algorithms. Data structures. Models of Computing (in Russian), Nizhny Novgorod State University Press, Nizhny Novgorod, 2006. [2] A. A. Andronov and L. S. Pontryagin, Rough systems (in Russian), Doklady Akademii nauk SSSR, 14 (1937), 247-250. [3] A. V. Bolsinov, S. V. Matveev and A. T. Fomenko, Topological classification of integrable Hamiltonian systems with two degrees of freedom. The list of systems of small complexity (in Russian), Uspekhi matematicheskikh nauk, 45 (1990), 49-77. doi: 10.1070/RM1990v045n02ABEH002344. [4] Yu. G. Borisovich, N. M. Bliznyakov, Ya. A. Izrailevich and T. N. Fomenko, Introduction to Topology (in Russian), "Vyssh. Shkola", Moscow, 1980. [5] A. Cobham, The intrinsic computational difficulty of functions, Proc. 1964 International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam, (1964), 24-30. [6] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman, San Francisco, 1979. [7] V. Grines, T. Medvedev and O. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Springer International Publishing Switzerland, 2016. [8] E. Ya. Gurevich and E. D. Kurenkov, Energy function and topological classification of Morse-Smale flows on surfaces (in Russian), Zhurnal SVMO, 17 (2015), 15-26. [9] D. König, Grafok es matrixok, Matematikai es Fizikai Lapok, 38 (1931), 116-119. [10] V. E. Kruglov, D. S. Malyshev and O. V. Pochinka, Multicolour graph as a complete topological invariant for $Ω$-stable flows without periodic trajectories on surfaces (in Russian), Matematicheskiy Sbornik, 209 (2018), 100-126. doi: 10.4213/sm8797. [11] V. E. Kruglov, T. M. Mitryakova and O. V. Pochinka, About types of cells of $Ω$ -stable flows without periodic trajectories on surfaces (in Russian), Dinamicheskie sistemy, 5 (2015), 43-49. [12] E. A. Leontovich and A. G. Mayer, About trajectories determining qualitative structure of sphere partition into trajectories (in Russian), Doklady Akademii Nauk SSSR, 14 (1937), 251-257. [13] E. A. Leontovich and A. G. Mayer, About scheme determining topological structure of partition into trajectories (in Russian), Doklady Akademii Nauk SSSR, 103 (1955), 557-560. [14] A. G. Mayer, Rough transformations of a circle (in Russian), Uchionye zapiski GGU. Gor'kiy, publikatsii. GGU, 12 (1939), 215-229. [15] G. Miller, Isomorphism testing for graphs of bounded genus, Proceedings of the 12th Annual ACM Symposium on Theory of Computing, (1980), 225-235. doi: 10.1145/800141.804670. [16] D. Neumann and T. O'Brien, Global structure of continuous flows on 2-manifolds, J. DifF. Eq., 22 (1976), 89-110. doi: 10.1016/0022-0396(76)90006-1. [17] A. A. Oshemkov and V. V. Sharko, About classification of Morse-Smale flows on 2-manifolds (in Russian), Matematicheskiy sbornik, 189 (1998), 93-140. doi: 10.1070/SM1998v189n08ABEH000341. [18] J. Palis, On the $C^1$ omega-stability conjecture, Publ. Math. Inst. Hautes Études Sci., 66 (1988), 211-215. [19] J. Palis and W. De Melo, Geometric Theory Of Dynamical Systems: An Introduction, Transl. from the Portuguese by A. K. Manning, New York, Heidelberg, Berlin, Springer-Verlag, 1982. [20] M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2. [21] M. Peixoto, Structural stability on two-dimensional manifolds (a further remarks), Topology, 2 (1963), 179-180. doi: 10.1016/0040-9383(63)90032-6. [22] M. Peixoto, On the Classification of Flows on Two-Manifolds, Dynamical systems Proc. Symp. held at the Univ. of Bahia, Salvador, Brasil, 1971. [23] C. Pugh and M. Shub, The $Ω$-stability theorem for flows, Inven. Math., 11 (1970), 150-158. doi: 10.1007/BF01404608. [24] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, Ann Arbor, London, Tokyo, 1995. [25] S. Smale, Differentiable dynamical systems, Bull. Amer. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.

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##### References:
 [1] V. E. Alekseev and V. A. Talanov, Graphs and Algorithms. Data structures. Models of Computing (in Russian), Nizhny Novgorod State University Press, Nizhny Novgorod, 2006. [2] A. A. Andronov and L. S. Pontryagin, Rough systems (in Russian), Doklady Akademii nauk SSSR, 14 (1937), 247-250. [3] A. V. Bolsinov, S. V. Matveev and A. T. Fomenko, Topological classification of integrable Hamiltonian systems with two degrees of freedom. The list of systems of small complexity (in Russian), Uspekhi matematicheskikh nauk, 45 (1990), 49-77. doi: 10.1070/RM1990v045n02ABEH002344. [4] Yu. G. Borisovich, N. M. Bliznyakov, Ya. A. Izrailevich and T. N. Fomenko, Introduction to Topology (in Russian), "Vyssh. Shkola", Moscow, 1980. [5] A. Cobham, The intrinsic computational difficulty of functions, Proc. 1964 International Congress for Logic, Methodology, and Philosophy of Science, North-Holland, Amsterdam, (1964), 24-30. [6] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W. H. Freeman, San Francisco, 1979. [7] V. Grines, T. Medvedev and O. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Springer International Publishing Switzerland, 2016. [8] E. Ya. Gurevich and E. D. Kurenkov, Energy function and topological classification of Morse-Smale flows on surfaces (in Russian), Zhurnal SVMO, 17 (2015), 15-26. [9] D. König, Grafok es matrixok, Matematikai es Fizikai Lapok, 38 (1931), 116-119. [10] V. E. Kruglov, D. S. Malyshev and O. V. Pochinka, Multicolour graph as a complete topological invariant for $Ω$-stable flows without periodic trajectories on surfaces (in Russian), Matematicheskiy Sbornik, 209 (2018), 100-126. doi: 10.4213/sm8797. [11] V. E. Kruglov, T. M. Mitryakova and O. V. Pochinka, About types of cells of $Ω$ -stable flows without periodic trajectories on surfaces (in Russian), Dinamicheskie sistemy, 5 (2015), 43-49. [12] E. A. Leontovich and A. G. Mayer, About trajectories determining qualitative structure of sphere partition into trajectories (in Russian), Doklady Akademii Nauk SSSR, 14 (1937), 251-257. [13] E. A. Leontovich and A. G. Mayer, About scheme determining topological structure of partition into trajectories (in Russian), Doklady Akademii Nauk SSSR, 103 (1955), 557-560. [14] A. G. Mayer, Rough transformations of a circle (in Russian), Uchionye zapiski GGU. Gor'kiy, publikatsii. GGU, 12 (1939), 215-229. [15] G. Miller, Isomorphism testing for graphs of bounded genus, Proceedings of the 12th Annual ACM Symposium on Theory of Computing, (1980), 225-235. doi: 10.1145/800141.804670. [16] D. Neumann and T. O'Brien, Global structure of continuous flows on 2-manifolds, J. DifF. Eq., 22 (1976), 89-110. doi: 10.1016/0022-0396(76)90006-1. [17] A. A. Oshemkov and V. V. Sharko, About classification of Morse-Smale flows on 2-manifolds (in Russian), Matematicheskiy sbornik, 189 (1998), 93-140. doi: 10.1070/SM1998v189n08ABEH000341. [18] J. Palis, On the $C^1$ omega-stability conjecture, Publ. Math. Inst. Hautes Études Sci., 66 (1988), 211-215. [19] J. Palis and W. De Melo, Geometric Theory Of Dynamical Systems: An Introduction, Transl. from the Portuguese by A. K. Manning, New York, Heidelberg, Berlin, Springer-Verlag, 1982. [20] M. Peixoto, Structural stability on two-dimensional manifolds, Topology, 1 (1962), 101-120. doi: 10.1016/0040-9383(65)90018-2. [21] M. Peixoto, Structural stability on two-dimensional manifolds (a further remarks), Topology, 2 (1963), 179-180. doi: 10.1016/0040-9383(63)90032-6. [22] M. Peixoto, On the Classification of Flows on Two-Manifolds, Dynamical systems Proc. Symp. held at the Univ. of Bahia, Salvador, Brasil, 1971. [23] C. Pugh and M. Shub, The $Ω$-stability theorem for flows, Inven. Math., 11 (1970), 150-158. doi: 10.1007/BF01404608. [24] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, Ann Arbor, London, Tokyo, 1995. [25] S. Smale, Differentiable dynamical systems, Bull. Amer. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.
The case when $U_\mathfrak c$ is homeomorphic to a Möbius band
$\phi^t$ and $\Upsilon_{\phi^t}$
The cases of the consistent (leftward) and the inconsistent (rightward) orientation of boundary's connecting component of some $\mathcal E$-region
A polygonal region
An example of the flow $f^t$ together with the polygonal regions
An example of $f^t$ and its four-colour graph
Two flows from $G$ and their equipped graphs
Two examples of flows from $G$ differing only by orientation of the limit cycle between $\mathcal M$ and $\mathcal A$ and their equipped graphs
Two examples of flow from $G$ without $\mathcal A$- and $\mathcal M$-regions differing only by orientation of the limit cycle and their equipped graphs
$f^t$, $\Gamma_{\mathcal M}$ and $\Gamma^*_{{\mathcal M}}$
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