  A system of transformations is associated to a rational point on an elliptic
curve. The sequence entropy is connected to the canonical height, and in some
cases there is a canonically defined quotient system whose entropy is the
canonical height and for which the fibre entropy is accounted for by local
heights at primes of bad reduction. The proofs use transcendence theory and a
strong form of Siegel's theorem. We go on to extend these ideas to the morphic
heights of Call and Goldstine.
