Топологическая сопряженность градиентно-подобных потоков на поверхностях и эффективные алгоритмы ее различения

Современная математика. Фундаментальные направления. 2022. Т. 68. № 3. С. 467–487.

Kruglov V., Pochinka O.

Gradient-like flows on surfaces have simple dynamics, which inspired many mathematicians
to search for invariants of their topological equivalence. Under assumptions of different generality
on the class of gradient-like flows under consideration, such classical invariants as the Leontovich–
Mayer scheme, the Peixoto graph, the equipped Peixoto graph, the two-color Wang graph, the three-
color Oshemkov–Sharko graph, the Fleitas circular scheme, etc. were obtained. Thus, the problem of
classifying gradient-like flows on surfaces from the point of view of topological equivalence has been
solved in an exhaustive way. In recent works by Kruglov, Malyshev, and Pochinka, it was proved that for
gradient-like flows the topological equivalence classes coincide with the topological conjugacy classes.
The obtained result allows us to use any invariants of their equivalence for topological conjugacy of
gradient-like flows. The present study is a review of the results on topological conjugacy of gradient-like
flows on surfaces and efficient algorithms for its distinguishing, that is, algorithms whose running time
is limited by some polynomial on the length of the input information. Gradient-like flows on surfaces have simple dynamics, which inspired many mathematicians
to search for invariants of their topological equivalence. Under assumptions of different generality
on the class of gradient-like flows under consideration, such classical invariants as the Leontovich–
Mayer scheme, the Peixoto graph, the equipped Peixoto graph, the two-color Wang graph, the three-
color Oshemkov–Sharko graph, the Fleitas circular scheme, etc. were obtained. Thus, the problem of
classifying gradient-like flows on surfaces from the point of view of topological equivalence has been
solved in an exhaustive way. In recent works by Kruglov, Malyshev, and Pochinka, it was proved that for
gradient-like flows the topological equivalence classes coincide with the topological conjugacy classes.
The obtained result allows us to use any invariants of their equivalence for topological conjugacy of
gradient-like flows. The present study is a review of the results on topological conjugacy of gradient-like
flows on surfaces and efficient algorithms for its distinguishing, that is, algorithms whose running time
is limited by some polynomial on the length of the input information.

Keywords: эффективный алгоритм, efficient algorithm, gradient-like flow, градиентно-подобный поток

Publication based on the results of: Теория динамических систем и ее приложения (2023)