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Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere
25/12 Bolshaya Pecherskaya Ulitsa, Nizhny Novgorod, 603155, Russian Federation |
In the present paper, a solution to the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere is obtained. It is precisely shown that with respect to the stable isotopic connectedness relation there exists countable many of equivalence classes of such systems. 43 words.
References:
[1] |
V. S. Afraimovich and L. P. Shilnikov,
On some global bifurcations associated with the disappearance of a saddle-node fixed point, Doc. USSR Acad. Sci., 219 (1974), 1281-1284.
|
[2] |
V. S. Afraimovich and L. P. Shilnikov,
On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742.
|
[3] |
A. Andronov and L. Pontryagin,
Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250.
|
[4] |
A. Banyaga,
The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283.
doi: 10.1016/0040-9383(77)90009-X. |
[5] |
A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. |
[6] |
P. R. Blanchard,
Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Math. J., 47 (1980), 33-46.
doi: 10.1215/S0012-7094-80-04704-3. |
[7] |
S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. |
[8] |
Kh. Bonatti, V. Z. Grines, V. S. Medvedev and O. V. Pochinka,
Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices, Tr. Mat. Inst. Steklova, 256 (2007), 54-69.
doi: 10.1134/S0081543807010038. |
[9] |
G. Fleitas,
Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51.
doi: 10.1007/BF02584749. |
[10] |
J. Franks,
Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S., 158 (1971), 301-308.
doi: 10.1090/S0002-9947-1971-0283812-3. |
[11] |
N. Gourmelon,
A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203.
doi: 10.1017/etds.2014.101. |
[12] |
V. Grines, E. Gurevich, O. Pochinka and D. Malyshev, On topological classification of Morse-Smale diffeomorphisms on the sphere $S^n$, preprint, arXiv: 1911.10234. |
[13] |
V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev and O. V. Pochinka,
Global attractor and repeller of Morse–Smale diffeomorphisms, Proc. Steklov Inst. Math., 271 (2010), 103-124.
doi: 10.1134/S0081543810040097. |
[14] |
V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer, Cham, 2016.
doi: 10.1007/978-3-319-44847-3. |
[15] |
V. Z. Grines, O. V. Pochinka and S. Van Strien,
On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.
doi: 10.17323/1609-4514-2016-16-4-727-749. |
[16] |
F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. |
[17] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
doi: 10.1007/BFb0092042. |
[18] |
B. von Kerékjártó,
Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38.
doi: 10.1007/BF01463232. |
[19] |
V. I. Lukyanov and L. P. Shilnikov,
On some bifurcations of dynamical systems with homoclinic structures, Doc. USSR Academy of Sciences, 243 (1978), 26-29.
|
[20] |
A. G. Mayer,
Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229.
|
[21] |
T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka and E. V. Shadrina,
On a class of isotopic connectivity of gradient-like maps of the 2-sphere with saddles of negative orientation type, Russ. J. Nonlinear Dyn., 15 (2019), 199-211.
doi: 10.20537/nd190209. |
[22] |
J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965.
![]() ![]() |
[23] |
S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5–71. |
[24] |
S. Newhouse, J. Palis and F. Takens,
Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc., 82 (1976), 499-502.
doi: 10.1090/S0002-9904-1976-14073-6. |
[25] |
S. Newhouse and M. M. Peixoto, There is a simple arc joining any two Morse-Smale flows, in Trois Études en Dynamique Qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41. |
[26] |
E. V. Nozdrinova,
Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle, Russ. J. Nonlinear Dyn., 14 (2018), 543-551.
doi: 10.20537/nd180408. |
[27] |
E. Nozdrinova and O. Pochinka,
On the existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on the 2-sphere, J. Phys. Conf. Ser., 990 (2018), 1-7.
doi: 10.1088/1742-6596/990/1/012010. |
[28] |
J. Palis and C. Pugh, Fifty problems in dynamical systems, in Dynamical Systems, Lecture Notes in Math., 468, Springer, Berlin, 1975,345–353.
doi: 10.1007/BFb0082633. |
[29] |
D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. |
[30] |
S. Smale,
Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626.
doi: 10.2307/2033664. |
[31] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |
show all references
References:
[1] |
V. S. Afraimovich and L. P. Shilnikov,
On some global bifurcations associated with the disappearance of a saddle-node fixed point, Doc. USSR Acad. Sci., 219 (1974), 1281-1284.
|
[2] |
V. S. Afraimovich and L. P. Shilnikov,
On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742.
|
[3] |
A. Andronov and L. Pontryagin,
Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250.
|
[4] |
A. Banyaga,
The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283.
doi: 10.1016/0040-9383(77)90009-X. |
[5] |
A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. |
[6] |
P. R. Blanchard,
Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Math. J., 47 (1980), 33-46.
doi: 10.1215/S0012-7094-80-04704-3. |
[7] |
S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. |
[8] |
Kh. Bonatti, V. Z. Grines, V. S. Medvedev and O. V. Pochinka,
Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices, Tr. Mat. Inst. Steklova, 256 (2007), 54-69.
doi: 10.1134/S0081543807010038. |
[9] |
G. Fleitas,
Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51.
doi: 10.1007/BF02584749. |
[10] |
J. Franks,
Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S., 158 (1971), 301-308.
doi: 10.1090/S0002-9947-1971-0283812-3. |
[11] |
N. Gourmelon,
A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203.
doi: 10.1017/etds.2014.101. |
[12] |
V. Grines, E. Gurevich, O. Pochinka and D. Malyshev, On topological classification of Morse-Smale diffeomorphisms on the sphere $S^n$, preprint, arXiv: 1911.10234. |
[13] |
V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev and O. V. Pochinka,
Global attractor and repeller of Morse–Smale diffeomorphisms, Proc. Steklov Inst. Math., 271 (2010), 103-124.
doi: 10.1134/S0081543810040097. |
[14] |
V. Z. Grines, T. V. Medvedev and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, Developments in Mathematics, 46, Springer, Cham, 2016.
doi: 10.1007/978-3-319-44847-3. |
[15] |
V. Z. Grines, O. V. Pochinka and S. Van Strien,
On 2-diffeomorphisms with one-dimensional basic sets and a finite number of moduli, Mosc. Math. J., 16 (2016), 727-749.
doi: 10.17323/1609-4514-2016-16-4-727-749. |
[16] |
F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. |
[17] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
doi: 10.1007/BFb0092042. |
[18] |
B. von Kerékjártó,
Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38.
doi: 10.1007/BF01463232. |
[19] |
V. I. Lukyanov and L. P. Shilnikov,
On some bifurcations of dynamical systems with homoclinic structures, Doc. USSR Academy of Sciences, 243 (1978), 26-29.
|
[20] |
A. G. Mayer,
Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229.
|
[21] |
T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka and E. V. Shadrina,
On a class of isotopic connectivity of gradient-like maps of the 2-sphere with saddles of negative orientation type, Russ. J. Nonlinear Dyn., 15 (2019), 199-211.
doi: 10.20537/nd190209. |
[22] |
J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965.
![]() ![]() |
[23] |
S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 57 (1983), 5–71. |
[24] |
S. Newhouse, J. Palis and F. Takens,
Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc., 82 (1976), 499-502.
doi: 10.1090/S0002-9904-1976-14073-6. |
[25] |
S. Newhouse and M. M. Peixoto, There is a simple arc joining any two Morse-Smale flows, in Trois Études en Dynamique Qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41. |
[26] |
E. V. Nozdrinova,
Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle, Russ. J. Nonlinear Dyn., 14 (2018), 543-551.
doi: 10.20537/nd180408. |
[27] |
E. Nozdrinova and O. Pochinka,
On the existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on the 2-sphere, J. Phys. Conf. Ser., 990 (2018), 1-7.
doi: 10.1088/1742-6596/990/1/012010. |
[28] |
J. Palis and C. Pugh, Fifty problems in dynamical systems, in Dynamical Systems, Lecture Notes in Math., 468, Springer, Berlin, 1975,345–353.
doi: 10.1007/BFb0082633. |
[29] |
D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. |
[30] |
S. Smale,
Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626.
doi: 10.2307/2033664. |
[31] |
S. Smale,
Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
doi: 10.1090/S0002-9904-1967-11798-1. |





















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