Rationality of dynamical canonical height

We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps, we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application of our results, we prove that if the canonical heights of two points a and b under the action of two rational functions f and g (defined over k) are positive rational numbers, and if the degrees of f and g are multiplicatively independent, then the orbit of a under f intersects the orbit of b under g in at most finitely many points, complementing the results of Ghioca-Tucker-Zieve.