Download PDF - Dynamical systems arising from elliptic curvesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Dynamical systems arising from elliptic curves

Authors 1, 1, 1, 1

Affiliations

  1. School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

Abstract

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by the real division polynomial (although the archimedean component of the map is artificial). Finally, we set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics.

Bibliography

  1. R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319.
  2. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, ibid. 153 (1971), 401-414.
  3. V. Chothi, G. Everest and T. Ward, S-integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99-132.
  4. S. David, Minorations des formes linéaires de logarithmes elliptiques, Mem. Soc. Math. France 62 (1995).
  5. G. Everest and T. Ward, A dynamical interpretation of the global canonical height on an elliptic curve, Experiment. Math. 7 (1998), 305-316.
  6. G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
  7. L. Flatto, J. C. Lagarias and B. Poonen, The zeta function of the beta transformation, Ergodic Theory Dynam. Systems 14 (1994), 237-266.
  8. E. Hewitt and K. Ross, Abstract Harmonic Analysis, Springer, New York, 1963.
  9. F. Hofbauer, β-shifts have unique maximal measures, Monatsh. Math. 85 (1978), 189-198.
  10. D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), 411-419.
  11. W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416.
  12. W. Parry, Representations for real numbers, ibid. 15 (1964), 95-105.
  13. A. Rényi, Representations for real numbers and their ergodic properties, ibid. 8 (1957), 477-493.
  14. J. F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), 399-448.
  15. K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, Basel, 1995.
  16. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986.
  17. J. H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339-358.
  18. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, New York, 1994.
  19. A. P. Veselov, What is an integrable mapping?, in: What is Integrability?, V. E. Zakharov (ed.), Springer, New York, 1991, 251-272.
  20. A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys. 145 (1992), 181-193.
  21. P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.
  22. M. Ward, The law of repetition of primes in an elliptic divisibility sequence, Duke Math. J. 15 (1948), 941-946.
  23. M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74.
  24. T. Ward, The entropy of automorphisms of solenoidal groups, Master's thesis, Univ. of Warwick, 1986.
  25. A. Weil, Basic Number Theory, third ed., Springer, New York, 1974.
Colloquium Mathematicum
2000/84/85/1
Export to PBN, Opens in new tab
Pages:
95-107
Main language of publication
English
Received
1999-05-21
Accepted
1999-11-16
Published
2000
Exact and natural sciences