  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.
