Given a product X of locally compact rank one Hadamard spaces, we study asymptotic properties of certain discrete isometry groups Gamma of X. First we give a detailed description of the structure of the geometric limit set and relate it to the limit cone; moreover, we showthat the action of Gamma on a quotient of the regular geometric boundary of X is minimal and proximal. This is completely analogous to the case of Zariski dense discrete subgroups of semi-simple Lie groups acting on the associated symmetric space (compare [5]). In the second part of the paper we study the distribution of Gamma-orbit points in X. As a generalization of the critical exponent delta(Gamma) of Gamma we consider for any theta is an element of R->= 0(r), parallel to theta parallel to = 1, the exponential growth rate delta(theta)(Gamma) of the number of orbit points in X with prescribed "slope" theta. In analogy to Quint's result in [26] we show that the homogeneous extension psi(Gamma) to R->= 0(r) of delta(theta)(Gamma) as a function of theta is upper semi-continuous, concave and strictly positive in the relative interior of the intersection of the limit cone with the vector subspace of R-r it spans. This shows in particular that there exists a unique slope theta* for which delta(theta*)(Gamma) is maximal and equal to the critical exponent of Gamma. We notice that an interesting class of product spaces as above comes from the second alternative in the Rank Rigidity Theorem ([12, Theorem A]) for CAT(0)-cube complexes. Given a finite-dimensional CAT(0)-cube complex X and a group Gamma of automorphisms without fixed point in the geometric compactification of X, then either Gamma contains a rank one isometry or there exists a convex Gamma-invariant subcomplex of X which is a product of two unbounded cube subcomplexes; in the latter case one inductively gets a convex Gamma-invariant subcomplex of X which can be decomposed into a finite product of rank one Hadamard spaces.
