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March
2021, 41(3): 1101-1131.
doi: 10.3934/dcds.2020311
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.
Mathematics Subject Classification:
Primary: 37B25, 37B35, 37D05, 37D15.
Citation:
Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete and Continuous Dynamical Systems,
2021, 41
(3)
: 1101-1131.
doi: 10.3934/dcds.2020311
![]()
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. |
Figure 2.
Saddle-node bifurcation
Figure 3.
Strongly stable and unstable foliations
Figure 4.
Flip bifurcation
Figure 5.
Illustration to the proof of lemma 3.1
Figure 6.
An example of an attractor $ A_f $ and repeller $ R_f $ for a gradient-like diffeomorphism $ f $
Figure 7.
For the shown diffeomorphism, there are two ways to choose the pair $ A_f,\,R_f $ : 1) $ A_f = cl\,W^u_\sigma,\,R_f = \alpha $ and 2) $ A_f = \omega\cup f(\omega),\,R_f = cl\,W^s_\sigma $
Figure 8.
Illustration to the lemma 3.2
Figure 9.
Phase portrait of diffeomorphism $ \chi_m $
Figure 10.
Diffeomorphism $ \phi_{1, 3} $
Figure 11.
Illustration to the lemma 6.3, case 1)
Figure 12.
Illustration to the lemma 6.3, case 2)
Figure 15.
Curve $ C_\sigma $
Figure 16.
Curve $ C_{\tilde \sigma} $
Figure 17.
$ f\in H_{k,m} $
Figure 18.
$ f\in F_{1,3} $
Figure 19.
$ f \in {F_{1,3}}$
Figure 20.
ϕ1, 3
Figure 21.
Illustration to the lemma 9.1, the case 1)
Figure 22.
Illustration to the lemma 9.1, case 1)
Figure 23.
Illustration to the lemma 9.1, case 2)
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