Skip to main content
SpringerLink
Log in

Trace maps

  • Conference paper
Beyond Quasicrystals

Part of the book series: Centre de Physique des Houches ((LHWINTER,volume 3))

Abstract

In several problems arising in the quasicrystal theory one is faced with the problem of computing as simply as possible the traces of matrices defined by recursion using a substitution scheme. To be more specific, let us take an example.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. J.-P. ALLOUCHE and J. PEYRIERE Sur une formule de récurrence sur les traces de produits de matrices associés à certaines substitutions. C. R. Acad. Sc. Paris 302 (II) (1986), 1135 - 1136.

    MathSciNet  ADS  MATH  Google Scholar 

  2. Y. AVISHAI, D. BEREND, and D. GLAUBMAN Minimum-Dimension Trace Maps for Substitution Sequences. Phys. Rev. Lett. 72 (1994), 1842-1845.

    Google Scholar 

  3. Marc CULLER and Peter B. SHALEN Varieties of group representations and splittings of 3-manifolds. Ann. of Math. 117 (1983), 109 - 146.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. FRICKE and F. KLEIN Vorlesungen über die Theorie der automorphen Function en. B. G. Teubner, Leipzig 1897. Reprint: Johnson Reprint Corporation ( Academic Press ), New York 1965.

    Google Scholar 

  5. R. D. HOROWITZ Characters of Free Groups Represented in the Two-Dimensional Special Linear Group. Comm. Pure and Applied Math. XXV (1972), 635 - 649.

    Google Scholar 

  6. R. D. HOROWITZ Induced automorphisms on Fricke Characters of Free Groups. Trans. Amer. Math. Soc. 208 (1975), 41 - 50.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. KOHMOTO, L.P. KADANOFF, and C. TANG Localization Problem in One Dimension: Mapping, and Escape. Phys. Rev. Lett. 50 (1983), 1870 - 1872.

    Article  MathSciNet  ADS  Google Scholar 

  8. M. KOLAR and M. K. ALI Trace maps associated with general two-letter substitution rules. Phys. Rev. A 42 (1990), 7112 - 7124.

    Article  MathSciNet  ADS  Google Scholar 

  9. M. KOLAR and F. NORI Trace maps of general substitutional sequences. Phys. Rev. B 42 (1990), 1062 - 1065.

    Article  MathSciNet  ADS  Google Scholar 

  10. U. LERON Trace Identities and Polynomial Identities of n x n Matrices. J. of Algebra 42 (1976), 369 - 377.

    Article  MathSciNet  MATH  Google Scholar 

  11. W. MAGNUS Rings of Fricke Characters and Automorphism Groups of Free Groups. Math. Z. 170 (1980), 91 - 103.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. MAGNUS, A. KARRASS, and D. SOLITAR Combinatorial Group Theory. Interscience Publishers, New York 1966, and Dover, New York 1976.

    Google Scholar 

  13. B. H. NEUMANN Die Autornorphismengruppe der freien gruppen. Math. Ann. 107 (1933), 367 - 386.

    Article  MathSciNet  Google Scholar 

  14. J. NIELSEN Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzengenden. Math. Ann. 78 (1918), 385 - 397.

    Article  Google Scholar 

  15. J. NIELSEN Die Isomorphismen der freien Gruppen. Math. Ann. 91 (1924), 169 - 209.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. PEYRIER.E On the trace map for products of matrices associated with substitutive sequences. J. Stat. Phys. 62 (1991), 411 - 414.

    Article  ADS  Google Scholar 

  17. J. PEYRIERE, Z. X. WEN, and Z. Y. WEN Polynômes Associés aux Endomorphismes de Groupes Libres. L’ns. Math. 39 (1993), 153 - 175.

    MathSciNet  MATH  Google Scholar 

  18. C. PROCESI The Invariant Theory of n x n Matrices. Advances in Math. 19 (1976), 306 - 381.

    Article  MathSciNet  MATH  Google Scholar 

  19. No. 4 (in Russian); English translation: Math. USSR Izvestija 8 (1974), 727 - 760.

    Google Scholar 

  20. C. TRAINA Representation of the Trace Polynomial of Cyclically Reduced Words in a Free Group on Two Generators. Ph.D. Thesis, Polytechnic Institute of New York.

    Google Scholar 

  21. Proc. Amer. Math. Soc. 79 (1980), 369 - 372.

    Google Scholar 

  22. A. WHITTEMORE On Special Linear Characters of Free Groups of Rank n 4. Proc. Amer. Math. Soc. 40 (1973), 383 - 388.

    MathSciNet  MATH  Google Scholar 

  23. Z.-X. WEN Diverses études sur l’automaticité. Thèse, Université Paris-Sud 1991.

    Google Scholar 

  24. Z.-X. WEN Relations polynomiales entre les traces de produits de matrices. Comptes Rendus Acad. Sci. Paris, 318 Série I (1994), 99 - 104.

    Google Scholar 

  25. Z.-X. WEN and Z.-Y. WEN On the leading term and the degree of the polynomial trace mapping associated with a substitution. J. Stat. Phys. 75 (1994), 627 - 641.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathématiques, bât. 425, Université Paris-Sud, Unité associée au CNRS 757, 91405, Orsay Cedex, France

    J. Peyrière

Authors
  1. J. Peyrière
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Université Paris VII-Denis Diderot, France

    Françoise Axel

  2. CECN-CNRS, Vitry, France

    Denis Gratias

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Peyrière, J. (1995). Trace maps. In: Axel, F., Gratias, D. (eds) Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03130-8_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-03130-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59251-8

  • Online ISBN: 978-3-662-03130-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation