×

Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices. (English. Russian original) Zbl 1153.37340

Proc. Steklov Inst. Math. 256, 47-61 (2007); translation from Tr. Mat. Inst. Steklova 256, 54-69 (2007).
Summary: We study bifurcations of Morse-Smale diffeomorphisms under a change of the embedding of the separatrices of saddle periodic points in the ambient 3-manifold. The results obtained are based on the following statement proved in this paper: for the 3-sphere, the space of diffeomorphisms of North Pole-South Pole type endowed with the \(C ^{1}\) topology is connected. This statement is shown to be false in dimension 6.

MSC:

37D15 Morse-Smale systems
37G10 Bifurcations of singular points in dynamical systems
57M25 Knots and links in the \(3\)-sphere (MSC2010)
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Banyaga, ”Sur la structure du groupe des difféomorphismes qui preservent une forme symplectique,” Comment. Math. Helv. 53, 174–227 (1978). · Zbl 0393.58007
[2] G. R. Belitskii, Normal Forms, Invariants, and Local Mappings (Naukova Dumka, Kiev, 1979) [in Russian]. · Zbl 0418.22009
[3] C. Bonatti and V. Z. Grines, ”Knots as Topological Invariants for Gradient-like Diffeomorphisms of the Sphere S 3,” J. Dyn. Control Syst. 6(4), 579–602 (2000). · Zbl 0959.37017
[4] Ch. Bonatti, V. Grines, and O. Pochinka, ”On Existence of a Smooth Arc Joining ”North Pole-South Pole” Diffeomorphisms,” Prepubl. (Inst. Math. Bourgogne, 2006), http://math.u-bourgogne.fr/topologie/prepub/bifs_4.pdf
[5] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois ({\(\Gamma\)}4 = 0) (Springer, Berlin, 1968), Lect. Notes Math. 53. · Zbl 0164.24502
[6] R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (Princeton Univ. Press, Princeton, NJ, 1977), Ann. Math. Stud. 88. · Zbl 0361.57004
[7] F. Laudenbach, Topologie de la dimension trois: homotopie et isotopie (Centre Math. Ecole Polytech., Paris, 1974), Astérisque 12. · Zbl 0293.57004
[8] S. Matsumoto, ”There Are Two Isotopic Morse-Smale Diffeomorphisms Which Cannot Be Joined by Simple Arcs,” Invent. Math. 51, 1–7 (1979). · Zbl 0416.58015
[9] J. W. Milnor, ”On Manifolds Homeomorphic to the 7-Sphere,” Ann. Math. 64(2), 399–405 (1956). · Zbl 0072.18402
[10] J. W. Milnor, Lectures on the h-Cobordism Theorem (Princeton Univ. Press, Princeton, NJ, 1965; Mir, Moscow, 1969). · Zbl 0161.20302
[11] J. W. Milnor, Topology from the Differentiable Viewpoint (The Univ. Press of Virginia, Charlottesville, VA, 1965).
[12] S. Newhouse and M. M. Peixoto, ”There Is a Simple Arc Joining Any Two Morse-Smale Flows,” Astérisque 31, 15–41 (1976). · Zbl 0324.58012
[13] J. Palis, Jr. and W. de Melo, Geometric Theory of Dynamical Systems (Springer, New York, 1982; Mir, Moscow, 1986).
[14] J. Palis and C. C. Pugh, ”Fifty Problems in Dynamical Systems,” Lect. Notes Math. 468, 345–353 (1975). · Zbl 0304.58011
[15] D. Pixton, ”Wild Unstable Manifolds,” Topology 16(2), 167–172 (1977). · Zbl 0355.58004
[16] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics (World Sci., Singapore, 1998; Inst. Komp’yuternykh Issledovanii, Izhevsk, 2004), Part 1. · Zbl 0941.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.