Algebraic dynamics, difference fields and model theory

(Joint work with E. Hrushovski).
M. Baker proved in 2007 that given a endomorphism f of P^{^1} of degree >1
and defined over a function field K, either all points of P^{^1}(K) of canonical height 0
are preperiodic, or the endomorphism f descends to the field of constants of K.

Szpiro and Tucker asked about generalising the result to higher
dimensional varieties. A reformulation by Szpiro of the hypotheses
in terms of limited sets allowed model theory to get into the picture,
more precisely the model theory of difference fields. We were therefore
able to give the correct formulation of the descent result in higher
dimension, and prove it.

This is what I will explain in the talk:  connections between
algebraic dynamics and difference fields;  translation of the
problem; how model theory of difference fields intervenes.