Dynamical systems arising from elliptic curves

Details Dynamical systems arising from elliptic curves Dynamical systems arising from elliptic curves

1999-07-02

arxiv.org/abs/math/9907014v1

Colloquium Mathematicum 84-85, No. 1, 2000, 95-107

Mathematics

Dynamical Systems

14 pages

Scientific paper

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.

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