Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere

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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

Elena Nozdrinova and Olga Pochinka

25/12 Bolshaya Pecherskaya Ulitsa, Nizhny Novgorod, 603155, Russian Federation

Received April 2020 Revised June 2020 Published August 2020

Fund Project: This work was supported by the Russian Science Foundation (project 17-11-01041), except for the section 3 which is partially supported by Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant (ag. 075-15-2019-1931) and section 5, the evidence of which was supported by Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" (project 19-7-1-15-1)

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.

Keywords: Stable arc, flip, saddle-node, gradient-like diffeomorphism, two-dimensional sphere.

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 & Continuous Dynamical Systems - A, 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. Google Scholar
[2]	V. S. Afraimovich and L. P. Shilnikov, On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742. Google Scholar
[3]	A. Andronov and L. Pontryagin, Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250. Google Scholar
[4]	A. Banyaga, The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283. doi: 10.1016/0040-9383(77)90009-X. Google Scholar
[5]	A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. Google Scholar
[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. Google Scholar
[7]	S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. Google Scholar
[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. Google Scholar
[9]	G. Fleitas, Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51. doi: 10.1007/BF02584749. Google Scholar
[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. Google Scholar
[11]	N. Gourmelon, A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203. doi: 10.1017/etds.2014.101. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. Google Scholar
[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. Google Scholar
[18]	B. von Kerékjártó, Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38. doi: 10.1007/BF01463232. Google Scholar
[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. Google Scholar
[20]	A. G. Mayer, Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229. Google Scholar
[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. Google Scholar
[22]	J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. Google Scholar
[30]	S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. doi: 10.2307/2033664. Google Scholar
[31]	S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

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. Google Scholar
[2]	V. S. Afraimovich and L. P. Shilnikov, On small periodic disturbances of autonomous systems, Doc. USSR Acad. Sci., 214 (1974), 739-742. Google Scholar
[3]	A. Andronov and L. Pontryagin, Rough systems, Doklady Akademii Nauk SSSR, 14 (1937), 247-250. Google Scholar
[4]	A. Banyaga, The structure of the group of equivariant diffeomorphism, Topology, 16 (1977), 279-283. doi: 10.1016/0040-9383(77)90009-X. Google Scholar
[5]	A. N. Bezdenezhnykh and V. Z. Grines, Realization of gradient-like diffeomorphisms of two-dimensional manifolds, Differential Integral Equations, (1985), 33–37. Google Scholar
[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. Google Scholar
[7]	S. K. Boldsen, Different versions of mapping class groups of surfaces, preprint, arXiv: 0908.2221. Google Scholar
[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. Google Scholar
[9]	G. Fleitas, Replacing tangencies by saddle-nodes, Bol. Soc. Brasil. Mat., 8 (1977), 47-51. doi: 10.1007/BF02584749. Google Scholar
[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. Google Scholar
[11]	N. Gourmelon, A Franks' lemma that preserves invariant manifolds, Ergodic Theory Dynam. Systems, 36 (2016), 1167-1203. doi: 10.1017/etds.2014.101. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	F. Harary, Graph Theory, Addison-Wesley Publishing Co., Reading, Mass.-Menlo Park, Calif.-London, 1969. Google Scholar
[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. Google Scholar
[18]	B. von Kerékjártó, Uber die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., 80 (1919), 36-38. doi: 10.1007/BF01463232. Google Scholar
[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. Google Scholar
[20]	A. G. Mayer, Rough transformation of a circle into a circle, Scientific Notes - Gorky State Univ., 12 (1939), 215-229. Google Scholar
[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. Google Scholar
[22]	J. Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, NJ, 1965. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	D. Rolfsen, Knots and Links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Houston, TX, 1990. Google Scholar
[30]	S. Smale, Diffeomorphisms of the $2$-sphere, Proc. Amer. Math. Soc., 10 (1959), 621-626. doi: 10.2307/2033664. Google Scholar
[31]	S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

Figure 1. Conjugate arcs

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Figure 2. Saddle-node bifurcation

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Figure 3. Strongly stable and unstable foliations

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Figure 4. Flip bifurcation

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Figure 5. Illustration to the proof of lemma 3.1

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Figure 6. An example of an attractor $ A_f $ and repeller $ R_f $ for a gradient-like diffeomorphism $ f $

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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 $

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Figure 8. Illustration to the lemma 3.2

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Figure 9. Phase portrait of diffeomorphism $ \chi_m $

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Figure 10. Diffeomorphism $ \phi_{1, 3} $

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Figure 11. Illustration to the lemma 6.3, case 1)

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Figure 12. Illustration to the lemma 6.3, case 2)

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Figure 13. A diffeomorphism $ f\in G^+ $, for which $ m_f = 1 $, $ \mu_f = 3 $

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Figure 14. Transition from the diffeomorphism $ f\in G_m $ to the diffeomorphism $ g\in H_{m} $

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Figure 15. Curve $ C_\sigma $

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Figure 16. Curve $ C_{\tilde \sigma} $

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Figure 17.

$ f\in H_{k,m} $

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Figure 18.

$ f\in F_{1,3} $

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Figure 19.

$ f \in {F_{1,3}}$

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Figure 20. ϕ_{1, 3}

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Figure 21. Illustration to the lemma 9.1, the case 1)

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Figure 22. Illustration to the lemma 9.1, case 1)

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Figure 23. Illustration to the lemma 9.1, case 2)

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