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Transactions of the Moscow Mathematical Society

This journal, a translation of Trudy Moskovskogo Matematicheskogo Obshchestva, contains the results of original research in pure mathematics.

ISSN 1547-738X (online) ISSN 0077-1554 (print)

The 2020 MCQ for Transactions of the Moscow Mathematical Society is 0.74.

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Necessary and sufficient conditions for the topological conjugacy of 3-diffeomorphisms with heteroclinic tangencies
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by T. M. Mitryakova and O. V. Pochinka
Translated by: E. Khukhro
Trans. Moscow Math. Soc. 2016, 69-86
DOI: https://doi.org/10.1090/mosc/253
Published electronically: November 28, 2016
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Abstract:

In this paper we consider a class of three-dimensional diffeomorphisms that differ from gradient-like systems through the presence of heteroclinic tangencies. It is well known that such cascades are not structurally stable. However, here we find a complete system of topological invariants for a certain meaningful class of such diffeomorphisms.
References
  • A. A. Andronov and L. S. Pontryagin, Rough systems, Dokl. Akad. Nauk SSSR 14 (1937), no. 5, 247–250. (Russian)
  • A. N. Bezdenezhnykh and V. Z. Grines, Dynamic properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I, Methods of the qualitative theory of differential equations (Russian), Gor′kov. Gos. Univ., Gorki, 1984, pp. 22–38, 198 (Russian). MR 866686
  • A. N. Bezdenezhnykh and V. Z. Grines, Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II [translation of Methods of the qualitative theory of differential equations (Russian), 24–31, Gor′kov. Gos. Univ., Gorki, 1987; MR0987437 (90e:58076)], Selecta Math. Soviet. 11 (1992), no. 1, 13–17. Selected translations. MR 1155895
  • G. R. Belickiĭ, Normal′nye formy, invarianty i lokal′nye otobrazheniya, “Naukova Dumka”, Kiev, 1979 (Russian). MR 537763
  • C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, Topological classification of gradient-like diffeomorphisms on 3-manifolds, Topology 43 (2004), no. 2, 369–391. MR 2052968, DOI 10.1016/S0040-9383(03)00053-3
  • Kh. Bonatti, V. Z. Grines, and O. V. Pochinka, Classification of Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits on 3-manifolds, Tr. Mat. Inst. Steklova 250 (2005), no. Differ. Uravn. i Din. Sist., 5–53 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(250) (2005), 1–46. MR 2200906
  • V. Z. Grines and O. V. Pochinka, Morse-Smale cascades on 3-manifolds, Uspekhi Mat. Nauk 68 (2013), no. 1(409), 129–188 (Russian, with Russian summary); English transl., Russian Math. Surveys 68 (2013), no. 1, 117–173. MR 3088080, DOI 10.1070/rm2013v068n01abeh004823
  • V. Z. Grines and O. V. Pochinka, Introduction to the topological classification of cascades on manifolds of dimension two and three, Regular and Chaotic Dynamics, Izhevsk, 2011. (Russian)
  • V. Z. Grines and O. V. Pochinka, Necessary conditions of topological conjugacy for three-dimensional diffeomorphisms with heteroclinic tangencies, Dynamical Systems 3 (2013), no. 31, issue 3–4, 185–200.
  • E. Leontovič and A. Maĭer, On a scheme determining the topological structure of the separation of trajectories, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 557–560 (Russian). MR 0072305
  • W. de Melo, Moduli of stability of two-dimensional diffeomorphisms, Topology 19 (1980), no. 1, 9–21. MR 559473, DOI 10.1016/0040-9383(80)90028-2
  • W. de Melo and J. Palis, Moduli of stability for diffeomorphisms, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 318–339. MR 591192
  • W. de Melo and S. J. van Strien, Diffeomorphisms on surfaces with a finite number of moduli, Ergodic Theory Dynam. Systems 7 (1987), no. 3, 415–462. MR 912376, DOI 10.1017/S0143385700004120
  • T. M. Mitryakova and O. V. Pochinka, Classification of the simplest diffeomorphisms of the sphere $S^2$ with a modulus of stability, Sovrem. Mat. Prilozh. 54, Trudy Mezhdunarodnoi Konferentsii po Dinamicheskim Sistemam i Differentsial’nym Uravneniyam. Ch. 2 (2008), 99–112 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 158 (2009), no. 2, 261–275. MR 2677139, DOI 10.1007/s10958-009-9389-z
  • T. M. Mitryakova and O. V. Pochinka, On necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency, Tr. Mat. Inst. Steklova 270 (2010), no. Differentsial′nye Uravneniya i Dinamicheskie Sistemy, 198–219 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 270 (2010), no. 1, 194–215. MR 2768947, DOI 10.1134/S0081543810030156
  • T. M. Mitryakova and O. V. Pochinka, Towards the problem of classifying diffeomorphisms of surfaces with finitely many moduli of topological conjugacy, Nonlinear Dynamics 6 (2010), no. 1, 91–105. (Russian)
  • S. Newhouse, J. Palis, and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 5–71. MR 699057
  • J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Dynamical systems, Vol. III—Warsaw, Astérisque, No. 51, Soc. Math. France, Paris, 1978, pp. 335–346. MR 0494283
  • Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541
  • M. M. Peixoto, On the classification of flows on $2$-manifolds, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 389–419. MR 0334289
  • Stephen Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199–206. MR 133139, DOI 10.2307/1970311
  • Leonid P. Shilnikov, Andrey L. Shilnikov, Dmitry V. Turaev, and Leon O. Chua, Methods of qualitative theory in nonlinear dynamics. Part I, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 4, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. With the collaboration of Sergey Gonchenko (Sections 3.7 and 3.8), Oleg Sten′kin (Section 3.9 and Appendix A) and Mikhail Shashkov (Sections 6.1 and 6.2). MR 1691840, DOI 10.1142/9789812798596
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Bibliographic Information
  • T. M. Mitryakova
  • Affiliation: Department of Mechanics and Mathematics, Nizhnii Novgorod State University, Nizhmii Novgorod, 603950 Russian Federation
  • Email: tatiana.mitryakova@yandex.ru
  • O. V. Pochinka
  • Affiliation: National Research University Higher School of Economics
  • Email: olga-pochinka@yandex.ru
  • Published electronically: November 28, 2016
  • Additional Notes: This research was supported by the Programme for Basic Research of the National Research University Higher School of Economics in 2016 (project no. 98 “Topological methods in dynamics”), the Russian Science Foundation (project no. 15-01-03689-a), and the research work in accordance with the commission 2014/134 for performing state work in the sphere of scientific activity in the framework of the basic part of the state commission of the Ministry for Education and Science of Russia for 2014–2016 (Nizhnii Novgorod State University)
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Moscow Math. Soc. 2016, 69-86
  • MSC (2010): Primary 37D20
  • DOI: https://doi.org/10.1090/mosc/253
  • MathSciNet review: 3643965