Skip to main content
SpringerLink
Log in

New directions in algebraic dynamical systems

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

To Henk Broer, on the occasion of his 60th birthday

Abstract

The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.

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. Kasteleyn, P.W., The Statistics of Dimers on a Lattice, Physica, 1961, vol. 27, pp. 1209–1225.

    Article  Google Scholar 

  2. Temperley, H.N.V. and Fisher, M.E., Dimer Problem in Statistical Mechanics—an Exact Result, Philos. Mag (8)., 1961, vol. 6, pp. 1061–1063.

    Article  MATH  MathSciNet  Google Scholar 

  3. Smyth, C.J., On Measures of Polynomials in Several Variables, Bull. Austral. Math. Soc., 1981, vol. 23, no. 1, pp. 49–63.

    Article  MATH  MathSciNet  Google Scholar 

  4. Smyth, C.J., An Explicit Formula for the Mahler Measure of a Family of 3-variable Polynomials, J. Théor. Nombres Bordeaux, 2002, vol. 14, no. 2, pp. 683–700.

    MATH  MathSciNet  Google Scholar 

  5. Kontsevich, M. and Zagier, D., Periods, Mathematics Unlimited—2001 and Beyond, Berlin: Springer, 2001, pp. 771–808.

    Google Scholar 

  6. Deninger, C., Deligne Periods of Mixed Motives, K-theory and the Entropy of Certain Z n-actions, J. Amer. Math. Soc., 1997, vol. 10, no. 2, pp. 259–281.

    Article  MATH  MathSciNet  Google Scholar 

  7. Boyd, David W., Mahler’s Measure and Special Values of L-functions, Experiment. Math., 1998, vol. 7, no. 1, pp. 37–82.

    MATH  MathSciNet  Google Scholar 

  8. Rodriguez-Villegas, F., Identities Between Mahler Measures, Number theory for the millennium, III (Urbana, IL, 2000), Natick, MA: A K Peters, 2002, pp. 223–229.

    Google Scholar 

  9. Lind, D., Schmidt, K., and Ward, T., Mahler Measure and Entropy for Commuting Automorphisms of Compact Groups, Invent. Math., 1990, vol. 101, no. 3, pp. 593–629.

    Article  MATH  MathSciNet  Google Scholar 

  10. Burton, R. and Pemantle, R., Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings via Transfer-impedances, Ann. Probab., 1993, vol. 21, no. 3, pp. 1329–1371.

    Article  MATH  MathSciNet  Google Scholar 

  11. Onsager, L., Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev., 1944, vol. 65, nos 3–4, pp. 117–149.

    Article  MATH  MathSciNet  Google Scholar 

  12. Fan, C. and Wu, F.Y., General Lattice Model of Phase Transitions, Phys. Rev. B, 1970, vol. 2, no. 3, pp. 723–733.

    Article  Google Scholar 

  13. Wu, F.Y., Exactly Solved Models: A Journey in Statistical Mechanics. Selected Papers with Commentaries, World Sci., 2009.

  14. Temperley, H.N.V., Enumeration of Graphs on a Large Periodic Lattice, in Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973), London Math. Soc. Lecture Note Ser., No. 13., London: Cambridge Univ. Press,, 1974, pp. 155–159.

    Google Scholar 

  15. Dhar, D., Self-organized Critical State of Sandpile Automaton Models, Phys. Rev. Lett., 1990, vol. 64, no. 14, pp. 1613–1616.

    Article  MATH  MathSciNet  Google Scholar 

  16. Solomyak, R., On Coincidence of Entropies for Two Classes of Dynamical Systems, Ergodic Theory Dynam. Systems, 1998, vol. 18, no. 3, pp. 731–738.

    Article  MATH  MathSciNet  Google Scholar 

  17. Fisher, M.E., On the Dimer Solution of Planar Ising Models, J. Math. Phys., 1966, vol. 7, pp. 1776.

    Article  Google Scholar 

  18. Bak, P., Tang, C., and Wiesenfeld, K., Self-Organized Criticality: An Explanation of the 1/f Noise, Phys. Rev. Lett., 1987, vol. 59, no. 4, pp. 381–384.

    Article  MathSciNet  Google Scholar 

  19. Bak, P., Tang, C., and Wiesenfeld, K., Self-organized Criticality, Phys. Rev. A, 1988, vol. 38, no. 1, pp. 364–374.

    Article  MathSciNet  Google Scholar 

  20. Redig, F., Mathematical Aspects of the Abelian Sandpile Model, in Mathematical Statistical Physics, Lecture Notes of the Les Houches Summer School 2005 (Les Houches), Eds.: Bovier, A. et al., Amsterdam: Elsevier, 2006, pp. 657–728.

    Google Scholar 

  21. Einsiedler, M. and Schmidt, K., Markov Partitions and Homoclinic Points of Algebraic Zd-actions, Tr. Mat. Inst. Steklova, 1997, vol. 216, pp. 265–284 [Proc. Steklov Inst. Math., 1997, no. 1 (216), pp. 259–279].

    MathSciNet  Google Scholar 

  22. Schmidt, K., Quotients of l ∞(ℤ, ℤ) and Symbolic Covers of Toral Automorphisms, In Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, Amer. Math. Soc. Transl. Ser. 2, vol. 217, Providence, RI: AMS, 2006, pp. 223–246.

    Google Scholar 

  23. Schmidt, K., Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-shifts, Monatsh. Math., 2000, vol. 129, no. 1, pp. 37–61.

    Article  MATH  MathSciNet  Google Scholar 

  24. Sidorov, N., An Arithmetic Group Associated with a Pisot Unit, and its Symbolic-dynamical Representation, Acta Arith., 2002, vol. 101, no. 3, pp. 199–213.

    Article  MATH  MathSciNet  Google Scholar 

  25. Lind, D. and Schmidt, K., Homoclinic Points of Algebraic Z d-actions, J. Amer. Math. Soc., 1999, vol. 12, no. 4, pp. 953–980.

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmidt, K. and Verbitskiy, E., Abelian Sandpiles and the Harmonic Model, Comm. Math. Phys., 2009, vol. 292, no. 3, pp. 721–759.

    Article  MathSciNet  Google Scholar 

  27. Lind, D., Schmidt, K., and Verbitskiy, E., Entropy and Growth Rate of Periodic Points of Algebraic ℤd-actions, Cont. Math.,, 2010 (to appear).

  28. Lind, D., Schmidt, K., and Verbitskiy, E., Atoral Polynomials and Homoclinic Points, 2010 (work in progress).

  29. Kenyon, R., Lectures on Dimers, in Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Providence, RI: AMS, 2009, pp. 191–230.

    Google Scholar 

  30. Kenyon, R. and Okounkov, A., Planar Dimers and Harnack Curves, Duke Math. J., 2006, vol. 131, no. 3, pp. 499–524.

    Article  MATH  MathSciNet  Google Scholar 

  31. Kenyon, R., Okounkov, A., and Sheffield, S., Dimers and Amoebae, Ann. of Math. (2), 2006, vol. 163, no. 3, pp. 1019–1056.

    Article  MATH  MathSciNet  Google Scholar 

  32. Bacher, R., de la Harpe, P., and Nagnibeda, T., The Lattice of Integral Flows and the Lattice of Integral Cuts on a Finite Graph, Bull. Soc. Math. France, 1997, vol. 125, no. 2, pp. 167–198.

    MATH  MathSciNet  Google Scholar 

  33. Baker, M. and Norine, S., Riemann-Roch and Abel-Jacobi Theory on a Finite Graph, Adv. Math., 2007, vol. 215, no. 2, pp. 766–788

    Article  MATH  MathSciNet  Google Scholar 

  34. Musiker, G., The Critical Groups of a Family of Graphs and Elliptic Curves Over Finite Fields, J. Algebraic Combin., 2009, vol. 30, no. 2, pp. 255–276.

    Article  MATH  MathSciNet  Google Scholar 

  35. Häggström, O., A Subshift of Finite Type That is Equivalent to the Ising Model, Ergodic Theory Dynam. Systems, 1995, vol. 15, no. 3, pp. 543–556.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Vienna, Nordbergstraße 15, A-1090, Vienna, Austria

    Klaus Schmidt

  2. Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090, Vienna, Austria

    Klaus Schmidt

  3. Mathematical Institute, University of Leiden, PO Box 9512, 2300 RA, Leiden, The Netherlands

    Evgeny Verbitskiy

  4. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK, Groningen, The Netherlands

    Evgeny Verbitskiy

Authors
  1. Klaus Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evgeny Verbitskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Schmidt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schmidt, K., Verbitskiy, E. New directions in algebraic dynamical systems. Regul. Chaot. Dyn. 16, 79–89 (2011). https://doi.org/10.1134/S1560354710520072

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354710520072

MSC2010 numbers

Keywords

Search

Navigation