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On the solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a 2-sphere. (English. Russian original) Zbl 1447.37029
Russ. Math. Surv. 75, No. 2, 383-385 (2020); translation from Usp. Mat. Nauk 75, No. 2, 195-196 (2020).
From the text: The problem of the existence of an arc with an at most countable (a finite) number of bifurcations and connecting structurally stable systems (Morse-Smale systems) on manifolds was number 33 in the list of 50 Palis-Pugh problems [J. Palis and C. Pugh, Fifty problems on dynamical systems, Lect. Notes Math. 468, 345-353 (1975; Zbl 0304.58011)]. In this note we outline a solution to this problem for gradient-like diffeomorphisms of a two-dimensional sphere.
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C20 Generic properties, structural stability of dynamical systems
37B35 Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
37D15 Morse-Smale systems
37G10 Bifurcations of singular points in dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
Full Text: DOI
[1] J. Palis and C. C. Pugh 1975 Lecture Notes in Math. 468 Springer-Verlag, Berlin-New York 345-353
[2] S. Newhouse, J. Palis, and F. Takens 1983 Inst. Hautes Études Sci. Publ. Math.57 5-71 · Zbl 0518.58031
[3] E. V. Nozdrinova 2018 Нелинейная динам.14 4 543-551 · Zbl 1421.37017
[4] E. V. Nozdrinova 2018 Nelinein. Din.14 4 543-551
[5] P. R. Blanchard 1980 Duke Math. J.47 1 33-46 · Zbl 0457.58008
[6] B. von Kerékjártó 1919 Math. Ann.80 1 36-38 · JFM 47.0526.05
[7] А. Н. Безденежных, В. З. Гринес 1985 Дифференциальные и интегральные уравнения, Сб. науч. тр. Изд-во Горьковск. ун-та, Горький 33-37
[8] English transl. A. N. Bezdenezhnykh and V. Z. Grines 1992 Selecta Math. Soviet.11 1 19-23
[9] V. Z. Grines, T. V. Medvedev, and O. V. Pochinka 2016 Dynamical systems on 2- and 3-manifolds Springer, Cham xxvi+295 pp. · Zbl 1417.37018
[10] C. Bonatti, V. Z. Grines, V. S. Medvedev, and O. V. Pochinka 2007 Proc. Steklov Inst. Math.256 1 47-61 · Zbl 1153.37340
[11] E. V. Nozdrinova and O. V. Pochinka 2018 J. Phys. Conf. Ser.990 012010 7 pp.
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