2 2 N ov 1 99 9 CANONICAL HEIGHTS AND ENTROPY IN ARITHMETIC DYNAMICS

The height of an algebraic number in the sense of Diophantine geometry is known to be related to the entropy of an automorphism of a solenoid associated to the number. An el-liptic analogue is considered, which necessitates introducing a notion of entropy for sequences of transformations. A sequence of transformations are defined for which there is a canonical arithmetically defined quotient whose entropy is the canonical height, and in which the fibre entropy is accounted for by local heights at primes of singular reduction, yielding a dynamical interpretation of singular reduction. This system is related to local systems, whose entropy coincides with the local canonical height up to sign. The proofs use transcendence theory, a strong form of Siegel's theorem, and an elliptic analogue of Jensen's formula. These elliptic systems are based upon iteration of the duplication map; the ideas extend to morphisms of projective space, giving examples where the associated entropies coincide with the morphic heights of Call and Goldstine. In particular, the local morphic heights at infinity for polynomials are realized as integrals over an associated Julia set with respect to the maximal measure, giving an analogue of the Jensen formula in that setting also.