We get three basic results in algebraic dynamics: (1). We give the first algorithm to compute the dynamical degrees to arbitrary precision. (2). We prove that for a family of dominant rational self-maps, the dynamical degrees are lower semi-continuous with respect to the Zariski topology. This implies a conjecture of Call and Silverman. (3). We prove that the set of periodic points of a cohomologically hyperbolic rational self-map is Zariski dense. Moreover, we show that, after a large iterate, every degree sequence grows almost at a uniform rate. This property is not satisfied for general submultiplicative sequences. Finally, we prove the Kawaguchi-Silverman conjecture for a class of self-maps of projective surfaces including all the birational ones. In fact, for every dominant rational self-map, we find a family of recursive inequalities of some dynamically meaningful cycles. Our proofs are based on these inequalities.

43 pages