There is a well-understood connection
between polynomials and certain simple algebraic dynamical
systems. In this connection, the Mahler measure corresponds
to the topological entropy, Kronecker's Theorem relates
ergodicity to positivity of entropy, approximants to the
Mahler measure are related to growth rates of periodic
points, and Lehmer's problem is related to the existence
of algebraic models for Bernoulli shifts.
There are similar relationships for higher-dimensional
algebraic dynamical systems.
¶ We review this connection, and indicate a possible
analogous connection between the global canonical height
attached to points on elliptic curves and a
possible 'elliptic' dynamical system.