  Let X be a normal projective variety defined over an algebraically closed
field of arbitrary characteristic. We study the sequence of intermediate
degrees of the iterates of a dominant rational selfmap of X, recovering former
results by Dinh, Sibony [DS05b], and by Truong [Tru16].Precisely, we give a new
proof of the submultiplicativity properties of these degrees and of its
birational invariance. Our approach exploits intensively positivity properties
in the space of numerical cycles of arbitrary codimension. In particular, we
prove an algebraic version of an inequality first obtained by Xiao [Xia15] and
Popovici [Pop16], which generalizes Siu's inequality (see [Trap95]) to
algebraic cycles of arbitrary codimension. This allows us to show that the
degree of a map is controlled up to a uniform constant by the norm of its
action by pull-back on the space of numerical classes in X.
