  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.
