Skip to main content
SpringerLink
Log in

Exact upper bounds for the number of one-dimensional basic sets of surfaceA-diffeomorphisms

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

ForA-diffeomorphisms of orientable and nonorientable surfaces estimates of the maximal number of their one-dimensional basic sets are given. This number depends on both the topological type of the surface and geometric properties of basic sets. It is shown that these estimates cannot be improved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Anosov and S. Aranson et al., Dynamical systems with hyperbolic behavior. (Russian) In: Itogi Nauki i Tekhniki: Sovremennye Problemy Matematiki, Fundamentalnye Napravleniya, Dynamical Systems-9.VINITI, Moscow, 1991, Vol. 66, 1–248. (English translation:Encyclopedia of Math. Sci., Springer-Verlag, Berlin).

    Google Scholar 

  2. S. Kh. Aranson, Trajectories on nonorientable two-dimensional manifolds.Math. USSR Sb. 9 (1969).

  3. S. Aranson and V. Grines, Topological classification of cascades on closed two-dimensional manifolds.Russ. Math. Surv. 45 (1990), No. 1, 1–35.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Kh. Aranson and E. Zhuzhoma, On the structure of quasiminimal sets of foliations on surfaces.Russ. Acad. Sci. Sb. Math. 82 (1994), No. 2, 397–424.

    Article  MATH  MathSciNet  Google Scholar 

  5. V. Grines, On topological conjugacy of diffeomorphisms of two-dimensional manifold on one-dimensional orientable basic sets. (Russian) Part 1,Trudy Mosk. Mat. Obsch. 32 (1975), 35–60; Part, 2,Trudy Mosk. Mat. Obshch. 34 (1977), 243–252.

    MATH  MathSciNet  Google Scholar 

  6. V. Grines and H. Kalay, About topological equivalence of diffeomorphisms with non-trivial basic sets on two-dimensional manifolds. In: Methods of the qualitative theory of differential equations. (Russian)Gor'kii Math. Sb. 1988, 40–48.

  7. G. Levitt, Feuilletages des surfaces.These, Paris, 1983.

  8. H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms.Comment. Math. Helvetici 68 (1993), 289–307.

    MATH  MathSciNet  Google Scholar 

  9. R. V. Plykin, On the topology of basic sets of Smale diffeomorphisms. (Russian)Math. Sb. 84 (1971), 301–312.

    MATH  MathSciNet  Google Scholar 

  10. —, Sources and sinkds ofA-diffeomorphisms of surfaces. (Russian)Math. Sb. 94 (1974), 243–264.

    MATH  MathSciNet  Google Scholar 

  11. —, On the existence of attractive (repulsive) periodic points ofA-diffeomorphisms of the projective plane and the Klein bottle. (Russian)Uspekhi Mat. Nauk 32 (1977), 179.

    MATH  MathSciNet  Google Scholar 

  12. —, On the geometry of hyperbolic attractors of smooth cascades. (Russian)Uspekhi Mat. Nauk 39 (1984), 75–113.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. R. F. Williams, TheD A-map of Smale and structural stability.Proc. Berk. Symp. Pure. Math. 14 (1970), 329–334.

    MATH  Google Scholar 

  15. A. Yu. Zhirov, Enumeration of hyperbolic attractors on orientable surfaces and applications to pseudo-Anosov diffeomorphisms. (Russian)Dokl. Ross. Akad. Nauk 330 (1993), 683–686.

    MATH  MathSciNet  Google Scholar 

  16. —, Hyperbolic attractors of diffeomorphisms of orientable surfaces. (Russian) Part 1,Math. Sb. 186 (1994), 3–50; Part 2,Math. Sb. 185 (1994), 29–80; Part 3.Math. Sb. 186 (1995), 59–82.

    Google Scholar 

  17. A. Yu. Zhirov and R. V. Plykin, Correspondence between one-dimensional hyperbolic attractors of surface diffeomorphisms and generalized pseudo-Anosov diffeomorphisms. (Russian)Mat. Zametki 58 (1995), No. 1, 149–152.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sci. Research Inst. of Appl., Math. and Cybernet., 603005, Nizhny Novgorod, Russia

    S. Kh. Aranson

  2. Gagarin Airforce Academy, 141170, Monino, Moscow reg., Russia

    A. Yu. Zhirov

  3. Dept. of Math. and Mech., Univ. of Nizhny Novgorod, 603005, Russia

    E. V. Zhuzhoma

  4. Dept. of Phys. and Math. Sci., Inst. of Atomic Energetics, 249020, Obninsk, Russia

    R. V. Plykin

Authors
  1. S. Kh. Aranson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. V. Plykin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Yu. Zhirov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. V. Zhuzhoma
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research was partially supported by the Russian Foundation for Fundamental Research, Grants 96-01-00236 (for the first and fourth authors) and 96-01-00626 (for the second and third ones) and RFFI-INTAS grant 00418i96.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aranson, S.K., Plykin, R.V., Zhirov, A.Y. et al. Exact upper bounds for the number of one-dimensional basic sets of surfaceA-diffeomorphisms. Journal of Dynamical and Control Systems 3, 1–18 (1997). https://doi.org/10.1007/BF02471759

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02471759

1991 Mathematics Subject Classification

Key words and phrases

Search

Navigation