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# Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle

Received 05 November 2018; accepted 14 November 2018

2018, Vol. 14, no. 4, pp.  543-551

Author(s): Nozdrinova E. V.

The problem of the existence of a simple arc connecting two structurally stable systems on a closed manifold is included in the list of the fifty most important problems of dynamical systems. This problem was solved by S. Newhouse and M. Peixoto for Morse – Smale flows on an arbitrary closed manifold in 1980. As follows from the works of Sh. Matsumoto, P. Blanchard, V. Grines, E.Nozdrinova, and O.Pochinka, for the Morse – Smale cascades, obstructions to the existence of such an arc exist on closed manifolds of any dimension. In these works, necessary and sufficient conditions for belonging to the same simple isotopic class for gradient-like diffeomorphisms on a surface or a three-dimensional sphere were found. This article is the next step in this direction. Namely, the author has established that all orientation-reversing diffeomorphisms of a circle are in one component of a simple connection, whereas the simple isotopy class of an orientation-preserving transformation of a circle is completely determined by the Poincar´e rotation number.
Keywords: rotation number, simple arc
Citation: Nozdrinova E. V., Rotation Number as a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  543-551
DOI:10.20537/nd180408

## References

[1] Andronov, A. A. and Pontryagin, L. S., “Rough Systems”, Dokl. Akad. Nauk SSSR, 14:5 (1937), 247–250 (Russian)
[2] Blanchard, P. R., “Invariants of the NPT Isotopy Classes of Morse – Smale Diffeomorphisms of Surfaces”, Duke Math. J., 47:1 (1980), 33–46
[3] Grines, V.Ż. and Pochinka, O. V., “On the Simple Isotopy Class of a Source-Sink Diffeomorphism on the $3$-Sphere”, Math. Notes, 94:5–6 (2013), 862–875        ; Mat. Zametki, 94:6 (2013), 828–845 (Russian)
[4] Grines, V., Medvedev, T., and Pochinka, O., Dynamical Systems on $2$- and $3$-Manifolds, Dev. Math., 46, Springer, New York, 2016, xxvi, 295 pp.
[5] Maier, A. G., “Rough Transform Circle into a Circle”, Uchen. Zap. Gorkov. Gos. Univ., 1939, no. 12, 215–229 (Russian)
[6] Matsumoto, Sh., “There Are Two Isotopic Morse – Smale Diffeomorphisms Which Cannot Be Joined by Simple Arcs”, Invent. Math., 51:1 (1979), 1–7
[7] Newhouse, S., Palis, J., and Takens, F., “Bifurcations and Stability of Families of Diffeomorphisms”, Inst. Hautes Études Sci. Publ. Math., 1983, no. 57, 5–71
[8] Newhouse, S. and Peixoto, M. M., “There Is a Simple Arc Joining Any Two Morse – Smale Flows”, Trois études en dynamique qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41
[9] Pochinka, O. V., Nozdrinova, E. V., and Kolobianina, A. E., “Classification of Rough Transformations of a Circle from a Modern Point of View”, Zh. Srednevolzhsk. Mat. Obshch., 19:1 (2017), 1–10 (Russian)
[10] Pochinka, O., Nozdrinova, E., and Dolgonosova, A., “On the Obstructions to the Existence of a Simple Arc Joining the Multidimensional Morse – Smale Diffeomorphisms”, Dinam. Sist., 7(35):2 (2017), 103–111
[11] Palis, J. and Pugh, C. C., “Fifty Problems in Dynamical Systems”, Dynamical Systems: Proc. Sympos. Appl. Topology and Dynamical Systems:Presented to E. C. Zeeman on His Fiftieth Birthday (Univ. Warwick, Coventry, 1973/1974), Lecture Notes in Math., 468, ed. A. Manning, Springer, Berlin, 1975, 345–353