Geodesic
Iterations of homogeneous polynomial transformations with positive coefficients
Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 411-416.

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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. 
@article{MZM_1969_6_4_a5,
     author = {I. V. Evstigneev},
     title = {Iterations of homogeneous polynomial transformations with positive coefficients},
     journal = {Matemati\v{c}eskie zametki},
     pages = {411--416},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a5/}
}
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I. V. Evstigneev. Iterations of homogeneous polynomial transformations with positive coefficients. Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 411-416. http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a5/