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

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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

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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

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