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Growth-complexity spectrum of some discrete dynamical systems. (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 3×3 matricies and consider the degree d(n) 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(n)=λ n , where λ 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:
37F20Combinatorics and topology
37J40Perturbations, normal forms, small divisors, KAM theory, Arnol’d diffusion