The construction of an energy function for three-dimensional cascades with a two-dimensional expanding attractor
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V. Z. Grines, M. K. Noskova and O. V. Pochinka
Translated by: E. Khukhro - Trans. Moscow Math. Soc. 2015, 237-249
- DOI: https://doi.org/10.1090/mosc/249
- Published electronically: November 18, 2015
Abstract:
In this paper we establish the existence of an energy function for structurally stable diffeomorphisms of closed three-dimensional manifolds whose nonwandering set contains a two-dimensional expanding attractor.References
- D. V. Anosov, Structurally stable systems, Trudy Mat. Inst. Steklov. 169 (1985), 59–93, 254 (Russian). Topology, ordinary differential equations, dynamical systems. MR 836569
- Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979–990. MR 27512, DOI 10.2307/1969408
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- V. Z. Grines, F. Laudenbakh, and O. V. Pochinka, A quasi-energy function for diffeomorphisms with wild separatrices, Mat. Zametki 86 (2009), no. 2, 175–183 (Russian, with Russian summary); English transl., Math. Notes 86 (2009), no. 1-2, 163–170. MR 2584553, DOI 10.1134/S0001434609070190
- V. Grines, F. Laudenbach, and O. Pochinka, Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds, Mosc. Math. J. 9 (2009), no. 4, 801–821, 935 (English, with English and Russian summaries). MR 2663991, DOI 10.17323/1609-4514-2009-9-4-801-821
- V. Z. Grines, F. Laudenbakh, and O. V. Pochinka, A dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds, Tr. Mat. Inst. Steklova 278 (2012), no. Differentsial′nye Uravneniya i Dinamicheskie Sistemy, 34–48 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 278 (2012), no. 1, 27–40. MR 3058781, DOI 10.1134/s0081543812060041
- V. Z. Grines and E. V. Zhuzhoma, Structurally stable diffeomorphisms with basic sets of codimension one, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 2, 3–66 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 2, 223–284. MR 1918843, DOI 10.1070/IM2002v066n02ABEH000378
- V. Z. Grines and O. V. Pochinka, An introduction to the topological classification of cascades on manifolds of dimension two and three, Regular Chaotic Dynam., Moscow–Izhevsk, 2011. (Russian)
- V. Grines and E. Zhuzhoma, On structurally stable diffeomorphisms with codimension one expanding attractors, Trans. Amer. Math. Soc. 357 (2005), no. 2, 617–667. MR 2095625, DOI 10.1090/S0002-9947-04-03460-9
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- Ricardo Mañé, A proof of the $C^1$ stability conjecture, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 161–210. MR 932138
- K. R. Meyer, Energy functions for Morse Smale systems, Amer. J. Math. 90 (1968), 1031–1040. MR 239220, DOI 10.2307/2373287
- J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
- Dennis Pixton, Wild unstable manifolds, Topology 16 (1977), no. 2, 167–172. MR 445552, DOI 10.1016/0040-9383(77)90014-3
- R. V. Plykin, The topology of basic sets of Smale diffeomorphisms, Mat. Sb. (N.S.) 84 (126) (1971), 301–312 (Russian). MR 0286134
- R. V. Plykin, Sources and sinks of $\textrm {A}$-diffeomorphisms of surfaces, Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336 (Russian). MR 0356137
- R. V. Plykin, Hyperbolic attractors of codimension one, Topology (Leningrad, 1982) Lecture Notes in Math., vol. 1060, Springer, Berlin, 1984, pp. 348–354. MR 770254, DOI 10.1007/BFb0099950
- M. M. Postnikov, Lectures on geometry. Semester V. Riemannian geometry, Factorial, Moscow, 1998. (Russian)
- Clark Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28–73. MR 474411, DOI 10.1016/0022-0396(76)90004-8
- Clark Robinson, Dynamical systems, 2nd ed., Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1999. Stability, symbolic dynamics, and chaos. MR 1792240
- Stephen Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199–206. MR 133139, DOI 10.2307/1970311
- S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 97–116. MR 165537
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- F. Wesley Wilson Jr. and James A. Yorke, Lyapunov functions and isolating blocks, J. Differential Equations 13 (1973), 106–123. MR 385251, DOI 10.1016/0022-0396(73)90034-X
- E. V. Zhuzhoma and V. S. Medvedev, On nonorientable two-dimensional basic sets on 3-manifolds, Mat. Sb. 193 (2002), no. 6, 83–104 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 869–888. MR 1957954, DOI 10.1070/SM2002v193n06ABEH000661
Bibliographic Information
- V. Z. Grines
- Affiliation: National Research University Higher School of Economics, Nizhniĭ Novgorod State University
- MR Author ID: 193726
- Email: vgrines@yandex.ru
- M. K. Noskova
- Affiliation: Nizhniĭ Novgorod State University
- Email: mknoskova@yandex.ru
- O. V. Pochinka
- Affiliation: National Research University Higher School of Economics, Nizhniĭ Novgorod State University
- Email: olga-pochinka@yandex.ru
- Published electronically: November 18, 2015
- Additional Notes: This research was supported by the Russian Foundation for Basic Research (grants no. 13-01-12452-ofi-m, 15-01-03687-a) and the Russian Science Foundation (grant no. 14-41-00044). This research paper uses the results of the project “Dynamical systems and their applications” carried out in the framework of the Programme of Basic Research of the National Research University Higher School of Economics in 2015.
- © Copyright 2015 V. Z. Grines, M. K. Noskova, O. V. Pochinka
- Journal: Trans. Moscow Math. Soc. 2015, 237-249
- MSC (2010): Primary 37D20
- DOI: https://doi.org/10.1090/mosc/249
- MathSciNet review: 3468066