Skip to main content
SpringerLink
Log in

Interval identification systems and plane sections of 3-periodic surfaces

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

Interval identification systems is a notion that, on the one hand, generalizes interval exchange transformations and, on the other hand, describes special cases of such transformations. In the present paper we overview some elementary facts, address a few questions about interval identification systems, and describe explicitly systems that allow one to construct 3-periodic surfaces in the 3-space whose intersections with planes of a fixed direction have chaotic behavior. The problem of asymptotic behavior of plane sections of 3-periodic surfaces was posed by S.P. Novikov in 1982 and studied then by his students. One of the most interesting remaining open questions about such sections is reduced to the study of interval identification systems.

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. M. Keane, “Interval Exchange Transformations,” Math. Z. 141, 25–31 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  2. W. A. Veech, “Gauss Measures for Transformations on the Space of Interval Exchange Maps,” Ann. Math., Ser. 2, 115(1), 201–242 (1982).

    MathSciNet  Google Scholar 

  3. H. Masur, “Interval Exchange Transformations and Measured Foliations,” Ann. Math., Ser. 2, 115(1), 169–200 (1982).

    Article  MathSciNet  Google Scholar 

  4. J. Hubbard and H. Masur, “Quadratic Differentials and Foliations,” Acta Math. 142(3–4), 221–274 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  5. I. Agol, J. Hass, and W. Thurston, “The Computational Complexity of Knot Genus and Spanning Area,” Trans. Am. Math. Soc. 358(9), 3821–3850 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Dynnikov and B. Wiest, “On the Complexity of Braids,” J. Eur. Math. Soc. 9(4), 801–840 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  7. S. P. Novikov, “The Hamiltonian Formalism and a Many-valued Analogue of Morse Theory,” Usp. Mat. Nauk 37(5), 3–49 (1982) [Russ. Math. Surv. 37 (5), 1–56 (1982)].

    Google Scholar 

  8. A. Ya. Maltsev and S. P. Novikov, “Dynamical Systems, Topology, and Conductivity in Normal Metals,” J. Stat. Phys. 115, 31–46 (2003); arXiv: cond-mat/0312708.

    Article  MathSciNet  Google Scholar 

  9. A. V. Zorich, “A Problem of Novikov on the Semiclassical Motion of an Electron in a Uniform Almost Rational Magnetic Field,” Usp. Mat. Nauk 39(5), 235–236 (1984) [Russ. Math. Surv. 39 (5), 287–288 (1984)].

    MathSciNet  Google Scholar 

  10. I. A. Dynnikov, “Semiclassical Motion of the Electron. A Proof of the Novikov Conjecture in General Position and Counterexamples,” in Solitons, Geometry, and Topology: On the Crossroad (Am. Math. Soc., Providence, RI, 1997), AMS Transl., Ser. 2, 179, pp. 45–73.

    Google Scholar 

  11. I. A. Dynnikov, “The Geometry of Stability Regions in Novikov’s Problem on the Semiclassical Motion of an Electron,” Usp. Mat. Nauk 54(1), 21–60 (1999) [Russ. Math. Surv. 54, 21–59 (1999)].

    MATH  MathSciNet  Google Scholar 

  12. G. Rauzy, “Echanges d’intervalles et transformations induites,” Acta Arith. 34, 315–328 (1979).

    MATH  MathSciNet  Google Scholar 

  13. R. De Leo and I. A. Dynnikov, “Geometry of Plane Sections of the Infinite Regular Skew Polyhedron {4, 6 | 4},” arXiv: 0804.1668.

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    I. A. Dynnikov

Authors
  1. I. A. Dynnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 72–84.

To S.P. Novikov on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dynnikov, I.A. Interval identification systems and plane sections of 3-periodic surfaces. Proc. Steklov Inst. Math. 263, 65–77 (2008). https://doi.org/10.1134/S0081543808040068

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543808040068

Keywords

Search

Navigation