General laws of the analytic linearization for random diffeomorphisms

Abstract

We show that random dynamical systems obey the similar laws of the linear analytic linearization as non-autonomous difference systems, which naturally cover the validity and invalidity of Poincaré and Seigel type theorems for random diffeomorphisms, respectively.

Uniform methods to stablish Poincaré type linearization theorems

“…Let f k (s, ω, t, y) = d l=1 f l k (s, ω, t)y k e l and g k (s, ω, t, y) = ad −1 Λ f k (s, ω, t, y). Therefore, still by formula (14) for a fixed k we have that…”

“…However, when we study the invalidity of Seigel type theorems for random diffeomorphisms, it is shown that there actually exist some common laws for general ones. Therefore, we extend the criterion [14,Theorem 3.4] to Theorem 1 for more systems, whose linear parts are near the diagonal constant ones. Then using it, we provide an extension of the results in [3] and new proofs of partial results in [15] and [16].…”