A dynamical interpretation of the global canonical height on an elliptic curve

There is a well-understood connection between polynomials and certain simpl e algebraic dynamical systems. In this connection, the Mahler measure corre sponds to the topological entropy, Kronecker's Theorem relates ergodicity t o positivity of entropy, approximants to the Mahler measure are related to growth rates of periodic points, and Lehmer's problem is related to the exi stence of algebraic models for Bernoulli shifts. There are similar relation ships for higher-dimensional algebraic dynamical systems. We review this connection, and indicate a possible analogous connection bet ween the global canonical height attached to points on elliptic curves and a possible 'elliptic' dynamical system.