Abstract
A method for constructing cascades on surfaces is developed, which makes it possible to model structurally unstable discrete dynamical systems with finitely many orbits of heteroclinic tangency and preset moduli of topological conjugacy.
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A. A. Andronov and L. S. Pontryagin, “Structurally stable systems,” Dokl. Akad. Nauk SSSR 14(5), 247–250 (1937).
E. A. Leontovich and A. G. Maier, “On a scheme determining the topological structure of the partition into orbits,” Dokl. Akad. Nauk SSSR 103(4), 557–560 (1955).
M. M. Peixoto, “On the classification of flows on 2-manifolds,” in Dynamical Systems (Academic, New York, 1973), pp. 389–419.
A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. I,” in Methods of the Qualitative Theory of Differential Equations (Izd. Gor’kovskogo Univ., Gorkii, 1984), pp. 22–38 [Sel. Math. Sov. 11 (1), 1–11 (1992)].
A. N. Bezdenezhnykh and V. Z. Grines, “Dynamical properties and topological classification of gradient-like diffeomorphisms on two-dimensional manifolds. II,” in Methods of the Qualitative Theory of Differential Equations (Izd. Gor’kovskogo Univ., Gorkii, 1987), pp. 24–31 [Sel. Math. Sov. 11 (1), 13–17 (1992)].
A. N. Bezdenezhnykh and V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds,” in Differential and Integral Equations (Izd. Gor’kovskogo Univ., Gorkii, 1985), pp. 33–37 [Sel. Math. Sov. 11 (1), 19–23 (1992)].
V. Z. Grines, “Topological classification of Morse-Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces,” Mat. Zametki 54(3), 3–17 (1993) [Math. Notes 54 (3), 881–889 (1993)].
J. Palis, “Adifferentiable invariant of topological conjugacies and moduli of stability,” in Dynamical Systems, Astérisque 51, 335–346 (1978).
W. de Melo, “Moduli of stability of two-dimensional diffeomorphisms,” Topology 19(1), 9–21 (1980).
W. de Melo and J. Palis, “Moduli of stability for diffeomorphisms,” in Lecture Notes in Math., Vol. 819: Global Theory of Dynamical Systems (Springer-Verlag, Berlin, 1980) pp. 318–339.
W. de Melo and S. J. van Strien, “Diffeomorphisms on surfaces with a finite number of moduli,” Ergodic Theory Dynam. Systems 7(3), 415–462 (1987).
T. M. Mitryakova and O. V. Pochinka, “Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency,” in TrudyMat. Inst. Steklov, Vol. 270: Differential Equations and Dynamical Systems (Nauka, Moscow, 2010), pp. 198–219 [Proc. Steklov Inst. Math. 270, 194–215 (2010)].
T. M. Mitryakova and O. V. Pochinka, “To the question of classification of diffeomorphisms of surfaces with finitely many moduli of topological conjugacy,” Nelinein. Dinam. 6(1), 91–105 (2010).
D. Rolfsen, Knots and Links, inMath. Lecture Ser. (Publish or Perish, Houston, TX, 1990), Vol. 7.
C. Bonatti, V. Grines, V. Medvedev, and E. Pécou, “Topological classification of gradient-like diffeomorphisms on 3-manifolds,” Topology 43(2), 369–391 (2004).
C. Bonatti, V. Z. Grines, and O. V. Pochinka, “Classification of Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits on 3-manifolds,” in Trudy Mat. Inst. Steklov, Vol. 250: Differ. Uravn. Dinam. Sist. (Nauka, Moscow, 2005), pp. 5–53 [Proc. Steklov Inst. Math., No. 3 (250), 1–46 (2005)].
C. Kosniowski, A First Course in Algebraic Topology (Cambridge Univ. Press, Cambridge, 1980; Mir, Moscow, 1983).
D. Pixton, “Wild unstable manifolds,” Topology 16(2), 167–172 (1977).
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Original Russian Text © T. M. Mitryakova, O. V. Pochinka, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 902–919.
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Mitryakova, T.M., Pochinka, O.V. Realization of cascades on surfaces with finitely many moduli of topological conjugacy. Math Notes 93, 890–905 (2013). https://doi.org/10.1134/S0001434613050271
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DOI: https://doi.org/10.1134/S0001434613050271