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Structural stability and asymptotic behavior of invariant manifolds ofA-diffeomorphisms of surfaces

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Abstract

In this paper, the interrelation between structural stability of a diffeomorphismf of a two-dimensional smooth closed orientable manifoldM of genusg≥0 and asymptotic behavior of stable and unstable manifolds of points of one-dimensional basic sets is studied. For a manifoldM of genusg≥1 with the universal covering\(\overline M \) we study also the problem of deviation from geodesics on\(\overline M \) of preimages of stable and unstable manifolds of the points of exteriorly situated one-dimensional basic sets.

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References

  1. D. V. Anosov, Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane.J. Dynam. and Control Syst. 1 (1995), No. 1, 125–138.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. V. Anosov, On one class of invariant sets of smooth dynamical systems. (Russian)Proc. 5th Int. Conf. Nonlinear Oscillations, Vol. 2, Qualitative Methods.Inst. Math. Acad. Nauk SSSR (1970), 39–45.

  3. V.Z. Grines, On the topological conjugacy of diffeomorphisms of two-dimensional manifolds on one-dimensional orientable basic sets. Part I. (Russian)Trans. Moscow Math. Soc. 32 (1975), 31–56;34 (1977), 237–245.

    MathSciNet  Google Scholar 

  4. —, On the topological conjugacy of diffeomorphisms of two-dimensional manifolds on one-dimensional orientable basic sets. Part II. (Russian),Trans. Moscow Math. Soc. 34 (1977), 237–245.

    MathSciNet  Google Scholar 

  5. R. V. Plykin, On topology of basic sets of S. Smale diffeomorphisms. (Russian)Math. USSR Sb. 84 (1971), No. 2, 301–312.

    MATH  MathSciNet  Google Scholar 

  6. —, Sources and sinks ofA-diffeomorphisms of surfaces. (Russian)Math. USSR Sb. 94 (1974), No. 2, 243–264.

    MATH  MathSciNet  Google Scholar 

  7. —, On geometry of hyperbolic attractors of smooth cascades. (Russian)Uspekhi Mat. Nauk. 39 (1984), No. 6 (240) 75–113.

    MATH  MathSciNet  Google Scholar 

  8. R. S. Robinson and R. F. Williams, Finite stability is not generic.Proc. Symp., Univ. of Brasil, 1971.Acad. Press, New York, London, 1973, 451–462.

  9. S.Kh. Aranson and V.Z. Grines, Topological classification of flows on closed two-dimensional manifolds.Russ. Math. Surv. 41 (1986), No. 1, 183–208.

    Article  MATH  MathSciNet  Google Scholar 

  10. —, Flows on two-dimensional manifolds. (Russian) In: Smooth dynamical systems, Ch. 4, D. V. Anosov et al., Eds., Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamental'nye Napravleniya, Vol. 1, Dynamical Systems 1,VINITI, Moscow, 1985, 229–240. (English translation:Encyclopedia of Math. Sci., Vol. 1,Springer-Verlag, Berlin, etc.)

    Google Scholar 

  11. —, Topological classification of cascades on closed two-dimensional manifolds. (Russian)Uspekhi Mat. Nauk 45 (1990), No. 1, 3–32. (English translation:Russ. Math. Surv. 45 (1990), No. 1, 1–35).

    MATH  MathSciNet  Google Scholar 

  12. —, Cascades on surfaces. (Russian) In: Dynamical systems with hyperbolic behavior. Ch. 3, D.V. Anosov et al., Eds., Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Fundamental'nye Napravleniya, Vol. 66, Dynamical Systems 9,VINITI, Moscow, 1991, 229–240. (English translation:Encyclopedia of Math. Sci., Vol. 66,Springer-Verlag, Berlin, etc).

    Google Scholar 

  13. S.Kh. Aranson, V.Z. Grines, and E.V. Zhuzhoma, On the geometry and topology of flows and foliations on surfaces and Anosov's problem. (Russian)Math. USSR Sb. 186 (1995), No. 2, 26–66.

    MathSciNet  Google Scholar 

  14. V.I. Pupko, Non-self-intersecting curves on closed surfaces. (Russian)Dokl. Akad. Nauk USSR 177 (1967), No. 2, 272–274. (English translation:Sov. Math. Dokl. 8 (1967).)

    MATH  MathSciNet  Google Scholar 

  15. J. Franks, Anosov diffeomorphisms. In: Global Analysis, Proc. Symp. Pure Math.Am. Math. Soc. Providence, R.I. 14 (1970), 61–94.

  16. R. Bowen, Periodic points and measures for axiomA-diffeomorphisms.Trans. Am. Math. Soc. 154 (1971), 337–397.

    Article  MathSciNet  Google Scholar 

  17. V.Z. Grines and Kh. Kh. Kalai, The topological classification of basic sets without pairs of conjugate points ofA-diffeomorphisms of surfaces. (Russian)Gor'kii Gos. Univ., Dept. VINITI, 1988, No. 1137-B 88, 1–95.

  18. V.Z. Grines and Kh. Kh. Kalai, On the topological equivalence of diffeomorphisms with nontrivial basic sets on two-dimensional manifolds. (Russian) In: Trudy Mezhvuz. Temat. Nauchn. Sb., Metody Kachestvennoi Teorii Differents. Uravn. i Teorii Bifurkatsii, E.A. Leontovich-Andronov, Ed.,Gor'kii State Univ, 40–48.

  19. A. V. Zhirov, Hyperbolic attactors of diffeomorphisms of orientable surfaces. Part 1. Coding, classification and coverings.Math. USSR Sb. 185 (1994), No. 6, 3–50; Part 2. Enumeration and application to pseudo-Anosov diffeomorphisms.Math. USSR Sb. 185 (1994), No. 9, 29–80; Part 3. Algorithm and classification.,Math. USSR Sb. 186 (1995), No. 9, 59–82.

    MATH  MathSciNet  Google Scholar 

  20. S. Smale, Differentiable dynamical systems.Bull. Am. Math. Soc. 73 (1967), 747–817.

    MATH  MathSciNet  Google Scholar 

  21. —, Morse inequalities for dynamical system.Bull. Am. Math. Soc. 66 (1960), 43–49.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Dept. of Math., State Agricultural Academy, 603107, Nizhnii Novgorod, Russia

    V. Z. Grines

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Grines, V.Z. Structural stability and asymptotic behavior of invariant manifolds ofA-diffeomorphisms of surfaces. Journal of Dynamical and Control Systems 3, 91–110 (1997). https://doi.org/10.1007/BF02471763

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