  For a product of Hadamard spaces $X=X_1\times X_2$ on which some group
$G\subset Is(X_1)\times Is(X_2)$ acting, G. Link arXiv:1107.3755v1 introduced
the growth rate $\delta_{\theta}$ of slope $\theta$ to construct a
$G-$invariant $(b,\theta)$-density.
  First, we show that $\delta_{\theta}$ is continuous in the slope $\theta$ as
the above group action with some mild condition.
  Second, we give a negative answer to a question raised by G. Link in \cite{1}
in general. And further results about the question are discussed.
  In the end, we can extent the $(b,\theta)$-densities in some reasonable
condition.
