General laws of the analytic linearization for random diffeomorphisms (Q415483)

The authors study normal forms of the (discrete) random dynamical systems generated by tempered analytic diffeomorphisms of the forms \[ \psi(\omega,x) = A(\omega)x + F(\omega,x) \] where \(0\) is a fixed point of \(\psi(\omega,.)\), \(A(\omega)=D\psi(\omega,0)\in Gl(d,\mathbb{C})\) and \(x\mapsto F(\omega,x)\) is analytic in some tempered ball for a.e. \(\omega\). Under some convenient conditions, the author proves essentially that \(\psi\) is locally analytically conjugated to its linear part \(A\). The obtained results cover, in particular, the validity and invalidity of Poincaré and Siegel type results for random diffeomorphisms, respectively.