Geodesic
Construction of energetic functions for $\Omega$-stable diffeomorphisms on $2$- and $3$-manifolds
Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 191-222 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we review the results connected to existence of the energetic function for discrete dynamical systems. Also we consider technique of construction of such functions for some classes of $\Omega$-stable and structurally stable diffeomorphisms on manifolds of dimension $2$ and $3$. 
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V. Z. Grines; O. V. Pochinka. Construction of energetic functions for $\Omega$-stable diffeomorphisms on $2$- and $3$-manifolds. Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 63 (2017) no. 2, pp. 191-222. http://geodesic.mathdoc.fr/item/CMFD_2017_63_2_a0/

[1] V. Z. Grines, “On topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers”, Mat. sb., 188:4 (1997), 57–94 (in Russian) | DOI | MR | Zbl

[2] V. Z. Grines, E. V. Zhuzhoma, “Structurally stable diffeomorphisms with basic sets of codimension one”, Izv. RAN. Ser. Mat., 66:2 (2002), 3–66 (in Russian) | DOI | MR | Zbl

[3] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms”, Tr. MIAN, 271, 2010, 111–133 (in Russian) | MR | Zbl

[4] V. Z. Grines, E. V. Zhuzhoma, O. V. Pochinka, “Rough diffeomorphisms with basic sets of codimension one”, Sovrem. mat. Fundam. napravl., 57, 2015, 5–30 (in Russian)

[5] V. Z. Grines, F. Laudenbakh, O. Pochinka, “Quasi-energy function for diffeomorphisms with wild separatrices”, Mat. zametki, 86:2 (2009), 175–183 (in Russian) | DOI | MR | Zbl

[6] V. Z. Grines, V. S. Medvedev, E. V. Zhuzhoma, “On surface attractors and repellers on 3-manifolds”, Mat. zametki, 78:6 (2005), 813–826 (in Russian) | DOI | MR | Zbl

[7] V. Z. Grines, M. K. Noskova, O. V. Pochinka, “Construction of the energy function for A-diffeomorphisms with two-dimensional nonwandering set on 3-manifolds”, Tr. Srednevolzhsk. Mat. ob-va, 17:3 (2015), 12–17 (in Russian) | Zbl

[8] V. Z. Grines, M. K. Noskova, O. V. Pochinka, “Construction of the energy function for three-dimensional cascades with two-dimensional expanding attractor”, Tr. Mosk. Mat. ob-va, 76, no. 2, 2015, 271–286 (in Russian) | Zbl

[9] V. S. Medvedev, E. V. Zhuzhoma, “On nonorientable two-dimensional basic sets on 3-manifolds”, Mat. sb., 193:6 (2002), 83–104 (in Russian) | DOI | MR | Zbl

[10] Dzh. Milnor, The Morse Theory, Platon, Volgograd, 1969 (in Russian) | MR

[11] T. M. Mitriakova, O. V. Pochinka, A. E. Shishenkova, “Energy function for diffeomorphisms of surfaces with finite hyperbolic chain recurrent set”, Zhurn. Srednevolzhsk. Mat. ob-va, 14:1 (2012), 98–107 (in Russian)

[12] A. A. Oshemkov, V. V. Sharko, “On classification of Morse–Smale flows on two-dimensional manifolds”, Mat. sb., 189:8 (1998), 93–140 (in Russian) | DOI | MR | Zbl

[13] R. V. Plykin, “Sources and sinks of A-diffeomorphisms of surfaces”, Mat. sb., 94(136):2 (1974), 243–264 (in Russian) | MR | Zbl

[14] R. V. Plykin, “On the structure of centralizers of Anosov diffeomorphisms of the torus”, Usp. mat. nauk, 53:6 (1998), 259–260 (in Russian) | DOI | MR | Zbl

[15] M. M. Postnikov, Lectures in Geometry. Semester V. Riemannian Geometry, Faktorial, Moscow, 1998 (in Russian)

[16] S. Smale, “Differentiable dynamical systems”, Usp. mat. nauk, 25:1 (1970), 113–185 (in Russian) | MR

[17] Artin E., Fox R. H., “Some wild cells and spheres in three-dimensional space”, Ann. Math., 49 (1948), 979–990 | DOI | MR | Zbl

[18] Bonatti Ch., Grines V., “Knots as topological invariant for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dyn. Control Syst., 6 (2000), 579–602 | DOI | MR | Zbl

[19] Conley C., Isolated invariant sets and Morse index, Am. Math. Soc., Providence, 1978 | MR | Zbl

[20] Debrunner H., Fox R., “A mildly wild imbedding of an $n$-frame”, Duke Math. J., 27 (1960), 425–429 | DOI | MR | Zbl

[21] Franks J., “Nonsingular Smale flow on $S^3$”, Topology, 24:3 (1985), 265–282 | DOI | MR | Zbl

[22] Franks J., “A variation on the Poincare–Birkhoff theorem”, Hamiltonian dynamical systems, Proc. AMS-INS-SIAM Jt. Summer Res. Conf., Contemporary Math., 81, 1988, 111–117 | DOI | MR | Zbl

[23] Grines V., Laudenbach F., Pochinka O., “Self-indexing energy function for Morse–Smale diffeomorphisms on 3-manifolds”, Mosc. Math. J., 9:4 (2009), 801–821 | MR | Zbl

[24] Grines V. Z., Laudenbach F., Pochinka O. V., “Dynamically ordered energy function for Morse–Smale diffeomorphisms on 3-manifolds”, Proc. Steklov Inst. Math., 278:1 (2012), 27–40 | DOI | MR | Zbl

[25] Grines V., Levchenko Y., Medvedev V., Pochinka O., “The topological classification of structural stable 3-diffeomorphisms with two-dimensional basic sets”, Nonlinearity, 28:11 (2015), 4081–4102 | DOI | MR | Zbl

[26] Grines V., Medvedev T., Pochinka O., Dynamical systems on 2- and 3-manifolds, Springer, Cham, 2016 | MR | Zbl

[27] Grines V., Zhuzhoma E., “On structurally stable diffeomorphisms with codimension one expanding attractors”, Trans. Am. Math. Soc., 357:2 (2005), 617–667 | DOI | MR | Zbl

[28] Harrold O. G., Griffith H. C., Posey E. E., “A characterization of tame curves in three-space”, Trans. Am. Math. Soc., 79 (1955), 12–34 | DOI | MR | Zbl

[29] Kaplan J., Mallet-Paret J., Yorke J., “The Lyapunov dimension of nowhere differentiable attracting torus”, Ergodic Theory Dynam. Systems, 2 (1984), 261–281 | MR

[30] Meyer K. R., “Energy functions for Morse–Smale systems”, Amer. J. Math., 90 (1968), 1031–1040 | DOI | MR | Zbl

[31] Palis J., “On Morse–Smale dynamical systems”, Topology, 8 (1969), 385–404 | DOI | MR

[32] Pixton D., “Wild unstable manifolds”, Topology, 16 (1977), 167–172 | DOI | MR | Zbl

[33] Robinson C., Dynamical Systems: stability, symbolic dynamics, and chaos, CRC Press, Boca Raton, 1999 | MR | Zbl

[34] Shub M., “Morse–Smale diffeomorphisms are unipotent on homology”, Dynamical Syst., Proc. Sympos. Univ. Bahia (Salvador, 1971), Academic Press, New York, 1973, 489–491 | MR

[35] Shub M., Sullivan D., “Homology theory and dynamical systems”, Topology, 4 (1975), 109–132 | DOI | MR

[36] Smale S., “Morse inequalities for a dynamical system”, Bull. Am. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[37] Smale S., “On gradient dynamical systems”, Annals Math., 74 (1961), 199–206 | DOI | MR | Zbl

[38] Smale S., “Differentiable dynamical systems”, Bull. Am. Math. Soc., 73 (1967), 747–817 | DOI | MR | Zbl

[39] Wilson W., “Smoothing derivatives of functions and applications”, Trans. Am. Math. Soc., 139 (1969), 413–428 | DOI | MR | Zbl