Normal form theory for volume preserving maps (Q1209753)

The normal form theory for analytic volume preserving maps on \(\mathbb{R}^ 3\) in the neighbourhood of the fixed point 0 is elaborated. The author studies the non-resonant case when \(\lambda^{m_ 1}_ 1\lambda^{m_ 2}_ 2\lambda^{m_ 3}_ 3=1\Leftrightarrow m_ 1= m_ 2=m_ 3\), where \(\lambda_ i\), \(i=1,2,3\), are eigenvalues of the linear part of the map. The main goal of the paper is to prove the asymptotic nature of the normal form series. This is achieved with the help of Cauchy's method.