Abstract
Let \({U \subset \mathbb{R}^{N}}\) be a neighbourhood of the origin and a function \({F:U\rightarrow U}\) 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 \({\varphi(F(x))=S\varphi(x)}\) , 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 \({2\leq\frac{log|s_1|}{log|s_N|} }\) 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
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 < j ≤ p.
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Zdun, M.C. On the Schröder equation and iterative sequences of C r diffeomorphisms in \({\mathbb{R}^{N}}\) space. Aequat. Math. 85, 1–15 (2013). https://doi.org/10.1007/s00010-012-0138-x
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DOI: https://doi.org/10.1007/s00010-012-0138-x