Skip to main content
SpringerLink
Log in

Perturbations of Nonhyperbolic Algebraic Automorphisms of the 2-Torus

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

All nonhyperbolic automorphisms of the 2-torus are not structurally stable, and it is generally impossible to predict the dynamics of their arbitrarily small perturbations. In this paper, given a representative of each algebraic conjugacy class of nonperiodic nonhyperbolic maps, a one-parameter family of diffeomorphisms is constructed, in which the zero value of the parameter corresponds to the given map and the nonzero values, to Morse–Smale diffeomorphisms. According to results of V. Z. Grines and A. N. Bezdenezhnykh, a Morse–Smale diffeomorphism of a closed orientable surface which induces a nonperiodic action on the fundamental group has nonempty heteroclinic set. It is proved that, in all of the constructed families, the diffeomorphisms corresponding to nonzero parameter values have nonempty orientable heteroclinic sets in which the number of orbits is determined by the automorphism being perturbed.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.

Similar content being viewed by others

Notes

  1. At the points \(z\in(k-1/2,k+1/2)\), \(k\in\mathbb Z\), the differentiability of this function follows from that of the functions \(\tan(\pi z)\) and \(\arctan(z)\); at the points \(k+1/2\), \(k\in\mathbb Z\), it follows from the coincidence of the left and right derivatives of the function at these points.

  2. Geometrically, this means that the graph of \(\phi_{\mathrm u}(x-(q+1))\) lies below that of \(\phi_{\mathrm u}(x-q)\), so that the point of intersection is below this graph as well (and it is on the right of it, because \(\phi_{\mathrm s}\) is monotone).

References

  1. A. N. Bezdenezhykh and V. Z. Grines, “Realization of gradient-like diffeomorphisms of two-dimensional manifolds,” Selecta Math. Soviet. 11 (1) (1992).

    Google Scholar 

  2. A. I. Morozov, “Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms,” Zhurnal SVMO 23 (2), 171–184 (2021).

    Article  MATH  Google Scholar 

  3. A. N. Bezdenezhnykh and V. Z. Grines, “Diffeomorphisms with orientable heteroclinic sets on two-dimensional manifolds,” in Methods of the Qualitative Theory of Differential Equations (Gorkovsk. Gos. Univ., Gorkii, 1985), pp. 139–152 [in Russian].

    Google Scholar 

  4. S. Kh. Aranson and V. Z. Grines, “The topological classification of cascades on closed two-dimensional manifolds,” Russian Math. Surveys 45 (1), 1–35 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, in Encyclopedia Math. Appl. (Cambridge Univ. Press, Cambridge, 1995), Vol. 54.

    Book  MATH  Google Scholar 

  6. S. Batterson, “The dynamics of Morse–Smale diffeomorphisms on the torus,” Trans. Amer. Math. Soc. 256, 395–403 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. V. Sidorov and E. E. Chilina, “On non-hyperbolic algebraic automorphisms of a two-dimensional torus,” Zhurnal SVMO 23 (3), 295–307 (2021).

    Article  MATH  Google Scholar 

  8. D. V. Anosov, “Geodesic flows on closed Riemannian manifolds of negative curvature,” Proc. Steklov Inst. Math. 90, 1–235 (1967).

    MathSciNet  Google Scholar 

  9. V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical Systems on 2- and 3-Manifolds, in Developments in Math. (Springer, Cham, 2016), Vol. 46.

    MATH  Google Scholar 

  10. V. Chigarev, A. Kazakov, and A. Pikovsky, “Kantorovich–Rubinstein–Wasserstein distance between overlapping attractor and repeller,” Chaos 30 (7), 073114 (2020).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The publication was prepared within the framework of the Academic Fund Program at the HSE University in 2021–2022 (grant no. 21-04-004), except for the work on Sec. 2, which was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-1101).

Author information

Authors and Affiliations

  1. HSE University, Nizhny Novgorod, 603155, Russia

    V. Z. Grines, D. I. Mints & E. E. Chilina

Authors
  1. V. Z. Grines
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. I. Mints
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. E. Chilina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. E. Chilina.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 229–243 https://doi.org/10.4213/mzm13612.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grines, V.Z., Mints, D.I. & Chilina, E.E. Perturbations of Nonhyperbolic Algebraic Automorphisms of the 2-Torus. Math Notes 114, 187–198 (2023). https://doi.org/10.1134/S0001434623070209

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434623070209

Keywords

Search

Navigation