Class of Stable Connectivity of Source-Sink Diffeomorphism on Two-Dimensional Sphere
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
CLASS OF STABLE CONNECTIVITY OF SOURCE-SINK DIFFEOMORPHISM ON TWO-DIMENSIONAL SPHERE E. V. Nozdrinova National Research University Higher School of Economics 25/12, Bol’shaya Pechorskaya St., Nizhny Novgorod 603155, Russia [email protected]
O. V. Pochinka ∗ National Research University Higher School of Economics 25/12, Bol’shaya Pechorskaya St., Nizhny Novgorod 603155, Russia [email protected]
UDC 517.9
We consider the class of gradient-like diffeomorphisms possessing an attractor and a repeller separated by a circle on a 2-sphere. For any diffeomorphism in this class we construct a stable arc connecting it with the source-sink system. Bibliography: 14 titles. Illustrations: 13 figures.
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Introduction
The notion of a stable arc connecting two structurally stable systems on a manifold was introduced in [1]. Such an arc does not change qualitative properties under small perturbations. As proved in [2], there exists a simple arc (containing only elementary bifurcations) between any two Morse–Smale flows. By the results of [3], such a simple arc can be always replaced with a stable arc. For Morse–Smale diffeomorphisms defined on manifolds of any dimension there are examples of systems that cannot be connected by a stable arc. Respectively, the following question naturally arises: find an invariant that uniquely determines the equivalence class of a Morse–Smale diffeomorphism with respect to the connection relation by a stable arc (a component of stable connection). A circle is a unique closed manifold for which this problem is completely solved. As shown in [4], for orientation-preserving rough transformations of a circle the component of stable connection is determined by the Poincar´e rotation number k/m, (k, m) = 1, while all orientationchanging diffeomorphisms lie in the same component of stable connection. For Morse–Smale diffeomorphisms on a two-dimensional sphere necessary conditions for the existence of a connecting stable arc were found in [5], where sufficient conditions were not ∗
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Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 85-98. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0094
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discussed. The conditions found in [5] imply that even on a two-dimensional sphere there are infinitely many components of stable connection. To clarify this fact, we regard S1 as the equator of the sphere S2 . Then the diffeomorphism of the circle with exactly two periodic orbits of period m and rotation number k/m can be extended to a diffeomorphism Fk/m : S2 → S2 possessing two fixed sources at the north and south poles. Moreover, the diffeomorphisms Fk/m and Fk /m for m = 2r · q and m = 2r · q , where r, r 0 are integers and q = q are natural numbers, are not connected by a stable arc (cf. Figure 1 for the phase portraits of diffeomorphisms of the 2-spheres F1 /2 and F1 /3).
(a)
(b)
Figure 1. Phase portraits of diffeomorphisms of 2-sphere (a) F1/2 , (b) F1/3 . The
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