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TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
The topological conjugacy criterion for surface Morse-Smale flows with a
finite number of moduli
Vladislav Kruglov, kruglovslava21@mail.ru
National Research University Higher School of Economics, Russian Federation
Coauthor: Olga Pochinka
The Morse-Smale flows have been classified in sense of topological equivalence for several
times during the last century. The most known invariants for such flows are invariants by E.
Leontovich and A. Mayer [1], [2], M. Peixoto [3], A. Oshemkov and V. Sharko [4]. Besides,
the Ω-stable flows on surfaces have been classified in such sense too by D. Neumann and T.
O’Brien [5] and V. Kruglov, D.Malyshev and O. Pochinka [6]. Attempts were also made to
classify Morse-Smale flows in sense of topological conjugacy: in particular, V. Kruglov [7]
proved that the classes of topological equivalence and topological conjugacy for gradient-like
flows on surfaces coincide.
J. Palis [8] considered a flow in a neighbourhood of a separatrix which connects two saddle
points. He showed that in each topological equivalence class there is continuum of topological
conjugacy classes, that is a flow with a separatrix-connection has analitical conjugacy invariants
called moduli of stability or moduli of topological conjugacy. Each limit cycle likewise gener-
ates at least one modulus associated with its period. V. Kruglov, O. Pochinka and G. Talanova
[9] proved that non-singular flows on an annulus with only two limit cycles on the annulus’s
boundary components have infinite number of moduli, expressed by a function.
The first result of this report is the following.
Theorem 1. A Morse-Smale surface flow has finite number of moduli iff it has no a trajectory
going from one limit cycle to another.
To construct the topological classification in sense of conjugacy we use the complete topo-
logical classifications in sense of equivalence from [4], [6]. Namely, there is one-to-one corre-
spondence between equivalent classes of a Morse-Smale flow φt on a surface and isomorphic
classes of the equipped graph Υφ∗t, which contains an information about partition the ambient
manifold into cells with similar trajectories behaviour and the limit cycles types.
To distinguish topological conjugacy classes we add to the equipped graph an information
on the periods of the limit cycles. It gives a new equipped graph Υφ∗∗t, and here is the second
result.
Theorem 2. Morse-Smale surface flows φt, φ t without trajectories going from one limit cycle to
another one are topologically conjugate iff their equipped graphs Υφ∗∗t and Υφ∗∗t are isomorphic.
Acknowledgements: The authors are partially supported by Laboratory of Dynamical Sys-
tems and Applications NRU HSE, grant No 075-15-2019-1931 of the Ministry of Science and
Higher Education of Russian Federation.
References
[1] Leontovich E. A., Mayer A. G. (1937). On the trajectories determining qualitative struc-
ture of sphere partition into trajectories (in Russian), Doklady Akademii nauk SSSR 14,
No.5, 251-257.
[2] Leontovich E. A., Mayer A. G. (1955). On the scheme determining topological structure of
partition into trajectories (in Russian), Doklady Akademii nauk SSSR 103, No.4, 557-560.
385
The topological conjugacy criterion for surface Morse-Smale flows with a
finite number of moduli
Vladislav Kruglov, kruglovslava21@mail.ru
National Research University Higher School of Economics, Russian Federation
Coauthor: Olga Pochinka
The Morse-Smale flows have been classified in sense of topological equivalence for several
times during the last century. The most known invariants for such flows are invariants by E.
Leontovich and A. Mayer [1], [2], M. Peixoto [3], A. Oshemkov and V. Sharko [4]. Besides,
the Ω-stable flows on surfaces have been classified in such sense too by D. Neumann and T.
O’Brien [5] and V. Kruglov, D.Malyshev and O. Pochinka [6]. Attempts were also made to
classify Morse-Smale flows in sense of topological conjugacy: in particular, V. Kruglov [7]
proved that the classes of topological equivalence and topological conjugacy for gradient-like
flows on surfaces coincide.
J. Palis [8] considered a flow in a neighbourhood of a separatrix which connects two saddle
points. He showed that in each topological equivalence class there is continuum of topological
conjugacy classes, that is a flow with a separatrix-connection has analitical conjugacy invariants
called moduli of stability or moduli of topological conjugacy. Each limit cycle likewise gener-
ates at least one modulus associated with its period. V. Kruglov, O. Pochinka and G. Talanova
[9] proved that non-singular flows on an annulus with only two limit cycles on the annulus’s
boundary components have infinite number of moduli, expressed by a function.
The first result of this report is the following.
Theorem 1. A Morse-Smale surface flow has finite number of moduli iff it has no a trajectory
going from one limit cycle to another.
To construct the topological classification in sense of conjugacy we use the complete topo-
logical classifications in sense of equivalence from [4], [6]. Namely, there is one-to-one corre-
spondence between equivalent classes of a Morse-Smale flow φt on a surface and isomorphic
classes of the equipped graph Υφ∗t, which contains an information about partition the ambient
manifold into cells with similar trajectories behaviour and the limit cycles types.
To distinguish topological conjugacy classes we add to the equipped graph an information
on the periods of the limit cycles. It gives a new equipped graph Υφ∗∗t, and here is the second
result.
Theorem 2. Morse-Smale surface flows φt, φ t without trajectories going from one limit cycle to
another one are topologically conjugate iff their equipped graphs Υφ∗∗t and Υφ∗∗t are isomorphic.
Acknowledgements: The authors are partially supported by Laboratory of Dynamical Sys-
tems and Applications NRU HSE, grant No 075-15-2019-1931 of the Ministry of Science and
Higher Education of Russian Federation.
References
[1] Leontovich E. A., Mayer A. G. (1937). On the trajectories determining qualitative struc-
ture of sphere partition into trajectories (in Russian), Doklady Akademii nauk SSSR 14,
No.5, 251-257.
[2] Leontovich E. A., Mayer A. G. (1955). On the scheme determining topological structure of
partition into trajectories (in Russian), Doklady Akademii nauk SSSR 103, No.4, 557-560.
385
