*(English)*Zbl 0940.37017

The authors study the iterations of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of the $3\times 3$ matricies and consider the degree $d\left(n\right)$ of the numerators (or denominators) of the corresponding successive rational expressions of the $n$-th iterate. The growth of this degree is (generically) exponential with $n$: $d\left(n\right)={\lambda}^{n}$, where $\lambda $ is a growth-complexity.

The seminumerical analysis which enables to compute this growth-complexity for all $9!$ possible birational transformations are presented. The generalizations of these results by replacing the permutations of the entries by homogeneous polynomial transformations of the entries are also given.

##### MSC:

37F20 | Combinatorics and topology |

37J40 | Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion |