Dynamical systems arising from elliptic curves

P. D'Ambros; G. Everest; R. Miles; T. Ward

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 95-107
  • ISSN: 0010-1354

Abstract

top
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.

How to cite

top

D'Ambros, P., et al. "Dynamical systems arising from elliptic curves." Colloquium Mathematicae 84/85.1 (2000): 95-107. <http://eudml.org/doc/210812>.

@article{DAmbros2000,
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.},
author = {D'Ambros, P., Everest, G., Miles, R., Ward, T.},
journal = {Colloquium Mathematicae},
keywords = {topological entropy; dynamical system; elliptic curve; global canonical height; adele},
language = {eng},
number = {1},
pages = {95-107},
title = {Dynamical systems arising from elliptic curves},
url = {http://eudml.org/doc/210812},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - D'Ambros, P.
AU - Everest, G.
AU - Miles, R.
AU - Ward, T.
TI - Dynamical systems arising from elliptic curves
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 95
EP - 107
AB - 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.
LA - eng
KW - topological entropy; dynamical system; elliptic curve; global canonical height; adele
UR - http://eudml.org/doc/210812
ER -

References

top
  1. [1] R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. Zbl0127.13102
  2. [2] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, ibid. 153 (1971), 401-414. Zbl0212.29201
  3. [3] V. Chothi, G. Everest and T. Ward, S-integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99-132. Zbl0879.58037
  4. [4] S. David, Minorations des formes linéaires de logarithmes elliptiques, Mem. Soc. Math. France 62 (1995). 
  5. [5] G. Everest and T. Ward, A dynamical interpretation of the global canonical height on an elliptic curve, Experiment. Math. 7 (1998), 305-316. Zbl0927.11009
  6. [6] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999. Zbl0919.11064
  7. [7] L. Flatto, J. C. Lagarias and B. Poonen, The zeta function of the beta transformation, Ergodic Theory Dynam. Systems 14 (1994), 237-266. Zbl0843.58106
  8. [8] E. Hewitt and K. Ross, Abstract Harmonic Analysis, Springer, New York, 1963. Zbl0115.10603
  9. [9] F. Hofbauer, β-shifts have unique maximal measures, Monatsh. Math. 85 (1978), 189-198. 
  10. [10] D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergodic Theory Dynam. Systems 8 (1988), 411-419. Zbl0634.22005
  11. [11] W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. Zbl0099.28103
  12. [12] W. Parry, Representations for real numbers, ibid. 15 (1964), 95-105. Zbl0136.35104
  13. [13] A. Rényi, Representations for real numbers and their ergodic properties, ibid. 8 (1957), 477-493. Zbl0079.08901
  14. [14] J. F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), 399-448. Zbl49.0712.02
  15. [15] K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, Basel, 1995. Zbl0833.28001
  16. [16] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, New York, 1986. Zbl0585.14026
  17. [17] J. H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), 339-358. Zbl0656.14016
  18. [18] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, New York, 1994. Zbl0911.14015
  19. [19] A. P. Veselov, What is an integrable mapping?, in: What is Integrability?, V. E. Zakharov (ed.), Springer, New York, 1991, 251-272. Zbl0733.58025
  20. [20] A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys. 145 (1992), 181-193. Zbl0751.58034
  21. [21] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. Zbl0475.28009
  22. [22] M. Ward, The law of repetition of primes in an elliptic divisibility sequence, Duke Math. J. 15 (1948), 941-946. Zbl0032.01403
  23. [23] M. Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74. Zbl0035.03702
  24. [24] T. Ward, The entropy of automorphisms of solenoidal groups, Master's thesis, Univ. of Warwick, 1986. 
  25. [25] A. Weil, Basic Number Theory, third ed., Springer, New York, 1974. 

NotesEmbed ?

top

You must be logged in to post comments.