Skip to main content
SpringerLink
Log in

A Lower Bound for Periods of Matrices

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

For a nonsingular integer matrix A, we study the growth of the order of A modulo N. We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values of N for which the order of A modulo N is logarithmically small. In contrast, we show that if the matrix is not exceptional, then the order of A modulo N goes to infinity faster than any constant multiple of  log N.

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. Ailon, N., Rudnick, Z.: Torsion points on curves and common divisors of ak - 1 and bk - 1. Acta Arith. 113 (1), 31–38 (2004)

    MATH  Google Scholar 

  2. Andreescu, T., Gelca, R.: Mathematical Olympiad challenges. Boston, MA: Birkhauser Boston, Inc., 2000

  3. Bugeaud, Y., Corvaja, P., Zannier, U.: An upper bound for the G.C.D of an-1 and bn-1. Math. Zeitschrift. 243, 79–84 (2003)

  4. Corvaja, P., Zannier, U.: A lower bound for the height of a rational function at S-unit points. Preprint, http://arxiv.org/abs/math.NT/0311030, 2003. To appear in Monatshefte für Math.

  5. Dyson, F.J., Falk, H.: Period of a discrete cat mapping. Am. Math. Monthly 99(7), 603–614 (1992)

    MATH  Google Scholar 

  6. Faure, F., Nonnenmacher, S., De Bievre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239, 449–492 (2003)

    MATH  Google Scholar 

  7. Hannay, J.H., Berry, M.V.: Quantization of linear maps on the torus-fresuel diffraction by a periodic grating. Phys. D 1(3), 267–290 (1980)

    Google Scholar 

  8. Keating, J.P.: Asymptotic properties of the periodic orbits of the cat maps. Nonlinearity 4(2), 277–307 (1991)

    Article  MATH  Google Scholar 

  9. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000)

    MATH  Google Scholar 

  10. Kurlberg, P., Rudnick, Z.: On Quantum Ergodicity for Linear Maps of the Torus. Commun. Math. Phys. 222(1), 201–227 (2001)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Mathematica e Inf, Via delle Scienze 206, 33100, Udine, Italy

    Pietro Corvaja

  2. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Zéev Rudnick

  3. I.U.A.V. -DCA, S. Croce 191, 30135, Venezia, Italy

    Umberto Zannier

Authors
  1. Pietro Corvaja
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zéev Rudnick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Umberto Zannier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pietro Corvaja.

Additional information

Communicated by P. Sarnak

Rights and permissions

Reprints and permissions

About this article

Cite this article

Corvaja, P., Rudnick, Z. & Zannier, U. A Lower Bound for Periods of Matrices. Commun. Math. Phys. 252, 535–541 (2004). https://doi.org/10.1007/s00220-004-1184-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1184-6

Keywords

Search

Navigation