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 Mat. Zametki, 2013, Volume 94, Issue 6, Pages 828–845 (Mi mz10363)

On the Simple Isotopy Class of a Source–Sink Diffeomorphism on the $3$-Sphere

V. Z. Grines, O. V. Pochinka

N. I. Lobachevski State University of Nizhni Novgorod

Abstract: The results obtained in this paper are related to the Palis–Pugh problem on the existence of an arc with finitely or countably many bifurcations which joins two Morse–Smale systems on a closed smooth manifold $M^n$. Newhouse and Peixoto showed that such an arc joining flows exists for any $n$ and, moreover, it is simple. However, there exist isotopic diffeomorphisms which cannot be joined by a simple arc. For $n=1$, this is related to the presence of the Poincaré rotation number, and for $n=2$, to the possible existence of periodic points of different periods and heteroclinic orbits. In this paper, for the dimension $n=3$, a new obstruction to the existence of a simple arc is revealed, which is related to the wild embedding of all separatrices of saddle points. Necessary and sufficient conditions for a Morse–Smale diffeomorphism on the $3$-sphere without heteroclinic intersections to be joined by a simple arc with a “source-sink” diffeomorphism are also found.

Keywords: isotopic diffeomorphisms, Morse–Smale diffeomorphism, source-sink diffeomorphism, wildly embedded separatrices, simple arc.

DOI: https://doi.org/10.4213/mzm10363

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English version:
Mathematical Notes, 2013, 94:6, 862–875

Bibliographic databases:

UDC: 517.938

Citation: V. Z. Grines, O. V. Pochinka, “On the Simple Isotopy Class of a Source–Sink Diffeomorphism on the $3$-Sphere”, Mat. Zametki, 94:6 (2013), 828–845; Math. Notes, 94:6 (2013), 862–875

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10363
• https://doi.org/10.4213/mzm10363
• http://mi.mathnet.ru/eng/mz/v94/i6/p828

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. Grines, O. Pochinka, E. Zhuzhoma, “On families of diffeomorphisms with bifurcations of attractive and repelling sets”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24:8 (2014), 1440015, 8 pp.
2. E. V. Nozdrinova, O. V. Pochinka, “On existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on 2-sphere”, European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018, UNSP 012010
3. V. Z. Grines, O. V. Pochinka, “Topological classification of global magnetic fields in the solar corona”, Dynam. Syst., 33:3 (2018), 536–546
4. E. Nozdrinova, “Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle”, Nelineinaya dinam., 14:4 (2018), 543–551
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