Rationality of dynamical canonical height

We present a dynamical proof of the well-known fact that the Néron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field , where is a curve. More generally, we investigate the mechanism by which the local canonical height for a map defined over a function field can take irrational values (at points in a local completion of ), providing examples in all degrees . Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math. 201 (2014), 631-700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights are rational and positive, for maps and of multiplicatively independent degrees and points , then the orbits and intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell-Lang conjecture. Invent. Math. 171 (2) (2008), 463-483].