Iterations of homogeneous polynomial transformations with positive coefficients

A study is made of the behavior as $k\to\infty$ of the iterations $T^k(x)$ of a homogeneous polynomial transformation $T$ acting from $R^n$ to $R^n$ according to the formula $(T(x))_i=Q_i(x)$, $i=1,2,\dots,n$, where $Q_i(x)$ is a homogeneous polynomial of degree $m>1$ with positive coefficients.