Projective surface automorphisms of positive topological entropy from an arithmetic viewpoint

Let X be a smooth projective surface over a number field K( subset of Q, and f: X -> X an automorphism of positive topological entropy. In this paper, we show that there are only finitely many f-periodic curves on X. Then we define a height function D corresponding to a certain nef and big R-divisor D on X and transforming well relative to f, and deduce some arithmetic properties off-periodic points and non f-periodic points.