# American Institute of Mathematical Sciences

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

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.

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

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##### 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
Conjugate arcs
Strongly stable and unstable foliations
Flip bifurcation
Illustration to the proof of lemma 3.1
An example of an attractor $A_f$ and repeller $R_f$ for a gradient-like diffeomorphism $f$
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) $A_f = \omega\cup f(\omega),\,R_f = cl\,W^s_\sigma$
Illustration to the lemma 3.2
Phase portrait of diffeomorphism $\chi_m$
Diffeomorphism $\phi_{1, 3}$
Illustration to the lemma 6.3, case 1)
Illustration to the lemma 6.3, case 2)
A diffeomorphism $f\in G^+$, for which $m_f = 1$, $\mu_f = 3$
Transition from the diffeomorphism $f\in G_m$ to the diffeomorphism $g\in H_{m}$
Curve $C_\sigma$
Curve $C_{\tilde \sigma}$
$f\in H_{k,m}$
$f\in F_{1,3}$
$f \in {F_{1,3}}$
ϕ1, 3
Illustration to the lemma 9.1, the case 1)
Illustration to the lemma 9.1, case 1)
Illustration to the lemma 9.1, case 2)
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