On the Schröder equation and iterative sequences of Cn n rn diffeomorphisms in {mathbb{R}^{N}} space

摘要

Let ({U subset mathbb{R}^{N}}) be a neighbourhood of the origin and a function ({F:Urightarrow U}) be of class C r , r≥ 2, F(0)=0. Denote by F n the n-th iterate of F and let ({0<|s_1|leq cdots leq|s_N| <1 }) , where ({s_1, ldots , s_N}) are the eigenvalues of dF(0). Assume that the Schröder equation ({varphi(F(x))=Svarphi(x)}) , where S:=dF(0) has a C 2 solution φ such that dφ(0)=id. If ({frac{log|s_1|}{log|s_N|} <2 }) then the sequence {S −n F n (x)} converges for every point x from the basin of attraction of F to a C 2 solution φ of (1). If ({2leqfrac{log|s_1|}{log|s_N|} }) then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S −n F n (x)}. Moreover, we show that if F is of class C r and ({r>big[frac{log|s_1|}{log|s_N|} big ]:=p geq 2}) then every C r solution of the Schröder equation such that dφ(0)=id is given by the formula$$begin{array}{ll}varphi (x)={limlimits_{n rightarrow infty}} (S^{-n}F^n(x) + {sumlimits _{k=2}^{p}} S^{-n}L_k (F^n(x))),end{array}$$where ({L_k:mathbb{R}^{N} rightarrow mathbb{R}^{N}}) are some homogeneous polynomials of degree k, which are determined by the differentials d (j) F(0) for 1