On the Schröder equation and iterative sequences of Cr diffeomorphisms in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^{N}}$$\end{document} space

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U \subset \mathbb{R}^{N}}$$\end{document} be a neighbourhood of the origin and a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F:U\rightarrow U}$$\end{document} be of class Cr, r ≥ 2, F(0) = 0. Denote by Fn the n-th iterate of F and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0<|s_1|\leq \cdots \leq|s_N| <1 }$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_1, \ldots , s_N}$$\end{document} are the eigenvalues of dF(0). Assume that the Schröder equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi(F(x))=S\varphi(x)}$$\end{document} , where S: = dF(0) has a C2 solution φ such that dφ(0) = id. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{log|s_1|}{log|s_N|} <2 }$$\end{document} then the sequence {S−nFn(x)} converges for every point x from the basin of attraction of F to a C2 solution φ of (1). If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2\leq\frac{log|s_1|}{log|s_N|} }$$\end{document} then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S−nFn(x)}. Moreover, we show that if F is of class Cr and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$$\end{document} then every Cr solution of the Schröder equation such that dφ(0) = id is given by the formula\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$$\end{document} are some homogeneous polynomials of degree k, which are determined by the differentials d(j)F(0) for 1 < j ≤ p.


Introduction
Let U be an open set in the N dimensional real space. The n-th iterate of a self-map F : U → U is defined by F n (x) = F n−1 (F (x)) and F 0 = id, the identity map. Assume that F is a C r -diffeomorphism, r ≥ 2, 0 ∈ U and 0 is the unique fixed point of F . Denote by d (k) F (0) the differential of F of k-th order at 0. The aim of the present paper is to investigate the convergence and divergence of the sequence {(dF (0)) −n F n (x)}. The limit of this sequence, if it exists, is strictly connected with C r solutions of the Schröder functional equation where ϕ : U → R N and S = dF (0), which is the fundamental equation of linearization of diffeomorphisms in the R N space. The idea goes back to Poincaré [16] and was further developed by Sternberg [19], Hartman [8], Chaperon [3], Kuczma [11] and [13]. In a one dimensional real case S ∈ R and it is easy to show that if 0 < |S| < 1 and Eq. (1) possesses a solution ϕ differentiable at 0 with ϕ (0) = 0 then (see for e.g., Szekeres [20], Kuczma [11] and for analytic functions Koenigs [9]). This limit exists also with some weaker assumptions on F (see e.g., Seneta [17], Dubic [6]). Intuitively the existence of the limit lim n→∞ F n (x) S n ∈ (0, ∞) may be interpreted as follows: the iterative sequence {F n (x)} tends to zero with the speed of convergence of the geometric sequence {S n }. We generalize this idea on the R N space. Another approach to the problem of the speed of convergence of the sequence {F n (x)} in a Banach space and its relation to the Schröder equation, where S is a real positive number, one can find in Walorski [21].
In this note we consider the following problem: does the above given formula but with matrix S still hold in the R N space? That is: if the equality ϕ(x) = lim n→∞ S −n F n (x) = lim n→∞ (dF (0)) −n F n (x) (2) is true for C r solutions of the Schröder equation? The answer is negative. The aim of this paper is to give some sufficient conditions for the truthfulness of formula (2) and also to give conditions which imply the falsehood of it. We also give a general formula expressing the diffeomorphic solution of Eq. (1). A similar problem but in the C N space was considered by Berteloot in [2]. He dealt with the mappings of the form F = M + p≥k H p , where H p are homogeneous polynomials in C N of degree p and M is an automorphism of C N . He showed that if k ≥ k 0 for a specially chosen k 0 then the sequence {M −n F n } converges to a locally biholomorphic solution ϕ of (1). Recall that the problem of holomorphic linearization in C N was initiated by Poincaré and Dulac in [16] and [7]. Since then numerous papers on this subject have been published and we refer to [1] for more details, some discussions and further references. We shall concentrate on the shape of diffeomorphic solutions of (1) in the R N space. Our approach to the problem is different from those which are useful for analytic functions. To obtain a desired formula we apply the ideas of Kuczma from [13]. Moreover, we assume for a given function F as minimal regularity assumptions as possible, which assure our formula. Vol. 85 (2013) On the Schröder equation and iterative sequences of C r 3

Auxiliary facts
Let r ∈ N and r ≥ 1. Assume the following general hypotheses: (i) U ⊂ R N is a neighbourhood of the origin and F : U → U is a C r diffeomorphism, F (0) = 0, |s 1 | ≤ · · · ≤ |s N | < 1 , where s 1 , s 2 , . . . , s N are the characteristic roots of the matrix S := dF (0).
If F of class C r with F (0) = 0 satisfies (H r ) we will say that F has no resonances.
Note that S is a nonsingular matrix and if ϕ is a solution of (1) in U then ϕ(0) = 0, since 1 is not an eigenvalue of S.
Suppose that the Schröder Eq. (1) has a C r solution such that dϕ(0) = E, where E is the identity matrix. Differentiating (1) k-times and setting x = 0 we get This means that the multidimensional matrices d (k) ϕ(0), k = 2, 3, . . . , r have to satisfy the system of Eq. (3). Define and . Hence we get the following Remark 1. The system (3) is equivalent to the system of equations where η is a polynomial function of degree r and the unknown quantities are multidimensional matrices d (k) η(0).
The elements of d (k) η(0) are N k -tuples of elements of R N and we can write for every permutation l 1 , . . . , l k of the sequence 1, . . . , k. We can write system (4) in the following form k s=1 N j1,...,js where β j1...js i1...i k are real numbers determined by the left hand side of equality (4) (for detailed evaluation see [13] p. 77). Systems . . , r with v 1 = E fulfilling equation (4) (or in other symbols equation (5)) and the consistency condition will be called admissible.
Obviously the admissible system of multidimensional matrices v k , k = 1, . . . , r determines uniquely the polynomial function η satisfying (4) This observation allows us to introduce the following where 2 ≤ deg L ≤ r or L ≡ 0 and η r satisfies (4) is said to be a formal solution of (1) of order r.
Note that, in general, a solution of system (4) may exist or not, and even if it exists, then there is no uniqueness attached to the solution. The existence and the uniqueness of a solution guarantees the condition for the lack of resonance (see Sternberg [19], Smajdor [18], Hartman [8] and Abate [1]). We can note this fact as follow Lemma 1. If F fulfils (i) and the characteristic roots s 1 , s 2 , . . . , s N of matrix S = dF (0) satisfy (H r ), then there exists a unique polynomial function η r : R N → R N of r-order satisfying system (4).
Note that every C r -solution ϕ of Eq. (1) determines uniquely a formal solution of r-order. This solution has the following form: We begin our considerations recalling the Kuczma generalization of a wellknown Sternberg and Hartman theorem on linearization (see [8,19]).
Denote by C r δ (U ) the set of all functions F : U → R N of class C r such that F (0) = 0 and Proposition 1 (Kuczma [12]). Suppose that F satisfies hypothesis (i) and

then for every system of admissible multidimensional matrices
. . , r and ϕ ∈ C r δ (V ). Let us note that if furthermore F has no resonance then Eq. (1) has a unique solution in the class C r δ (V ) such that dϕ(0) = E, where E is the identity matrix.
Definition. Every C r solution ϕ of (1) such that ϕ(0) = 0 and dϕ(0) = E is said to be a C r Schröder function of F.
Note that in the literature this name is used in various but similar meanings (see e.g., Dubuc [5,6]).

General results
We start with a theorem which gives a shape of C r solutions of Eq. (1). This is a continuation and application of Kuczma's ideas from paper [13].
If r = 1 then in a neighbourhood V of the origin there exists the limit If r ≥ 2 and (1) has a C r -solution γ = 0 then there exists the limit To prove our theorem we recall briefly the idea of M. Kuczma's proof of Proposition 3 from the paper [13] (see also the version for r ≥ 2 in [14] pp. 336-338).
Let ε > 0 be such that (|s N | + ε) r+δ (|s 1 | −1 + ε) < 1. By the Ostrowski lemma (see e.g. [14,15] p. 19 and [10], §4) there exists a nonsingular matrix A such that where || || denotes the Euclidean norm in the R N space. Hence we get . Thus we may assume that Consequently we can find a neighbourhood V := {x ∈ R N : ||x|| ≤ a}, a > 0 of the origin such that Let γ be a C r solution of (1) such that dγ(0) = id. Define the space (Ξ, ) for r ≥ 2 and In Ξ we introduce the metric (σ 1 , σ 2 ) := sup x ≤b where the norm v r is defined for multidimensional matrices in Using (7), (8) and repeating the same arguments as in [13] and [14] p. 337 we get that (Ξ, ) is a complete metric space and the transformation T given by the formula maps space Ξ into itself and satisfies the Lipschitz condition with a constant l < 1. Thus, by the Banach theorem, there exists a unique solution ϕ ∈ Ξ of (1) and this solution is given by the formula Vol. 85 (2013) On the Schröder equation and iterative sequences of C r 7 If r = 1 then σ 0 = id ∈ Ξ and consequently formula (2) . Hence we get formula (6). In general, id ∈ Ξ and even, for every n ∈ N, T n id = S −n F n ∈ Ξ. As a consequence of Theorem 1, Proposition 1 and Lemma 1 we get the following partial answer to our problem. Theorem 2. Let F of class C 2 satisfy hypothesis (i) and Then in a neighbourhood of the origin equation (1) has a unique C 2 solution ϕ such that ϕ(0) = 0 and dϕ(0) = E. It is given by the formula Proof. Note that the eigenvalues of dF (0) satisfy hypothesis (H 2 ). In fact (H 2 ) reads as follows s i s j = s k for all i, j, k = 1, . . . , N. Since |s 1 | ≤ |s i | ≤ |s N | we have |s i s j | ≤ |s N | 2 < |s 1 | ≤ |s k |. By Lemma 1 and Proposition 1 the Schröder Eq. (1) has a unique C 2 solution ϕ such that ϕ(0) = 0 and dϕ(0) = E. Since ϕ belongs to class C 1 1 it follows by Theorem 1, where r = 1 and δ = 1, that ϕ is given by formula (2).
In the case of holomorphic functions the similar thesis one can also deduce from the theorem of Berteloot mentioned in Sect.1 (see Th. 4.2 in [2]).
In the case r ≥ 3 as a consequence of Theorem 1 we obtain the following where L k (x) are polynomial functions of k-th order. Then there exists a unique C r solution ϕ of (1) such that d (k) ϕ(0) = d (k) η(0) for k = 1, . . . , r in a neighbourhood V of the origin and it is given by the formula Proof. It follows by Proposition 1 for δ = 0 that Eq. (1) has a unique C r -solu- we get, by Theorem 1, that ϕ is given by formula (9).
By Theorem 3 and Lemma 1 we get the following supplement to the Sternberg theorem ] such that dϕ(0) = E and ϕ is given by formula (9).

Particular results
If for r ≥ 3 we assume hypothesis (H r ) then the determination of the N -dimensional polynomials L k (x 1 , . . . , x N ) is very strenuous. In this section we consider the problem when the part of formula (9) with factors L k disappears and (9) has a simpler form (2). To this aim we shall deal with the asymptotic behavior of the sequence {S −n F n (x)} in the case, where r ≥ 3 and |s 1 | < |s N | 2 .
Remark 2. Let J be a Jordan form of matrix S and P be a nonsingular matrix such that P SP −1 = J. Then ϕ is a C r Schröder function of F if and only if In particular, if ϕ satisfies the assumptions of Theorem 3, then is a formal solution of the equation ψ(G(x)) = Jψ(x) of (r − 1)-order. Moreover, if ϕ is given by formula (9), then M k = P • L k • P −1 . Hence we may further assume that S has a Jordan form.
Assume that F satisfies the assumptions of Theorem 3 and the matrix dF (0) = S is diagonalizable. Thus, in view of Remark 2, we may assume that S has the diagonal form, i.e., where 0 < |s 1 | ≤ · · · ≤ |s N | < 1.

Vol. 85 (2013)
On the Schröder equation and iterative sequences of C r 9 Let ϕ be a C r solution of (1) such that ϕ(0) = 0 and dϕ(0) = E. By (9) we have in a neighbourhood of the origin V. Put We know that L k (x) is a homogenous polynomial of k-order. The jth-coordinate of L k (x) we can write in the following form where x = (x 1 , . . . , x N ). Since 10

M. C. Zdun AEM
On the other hand Hence we infer that the j-th coordinate of z n (x) is expressed by the formula (12) By (10) we have where The above mentioned observations let us to prove the following Proof. Fix j ≥ p. Note that By the assumption the sequences {y i1 n (x 0 ), . . . , y i k n (x 0 ))} are bounded, thus (11) implies that lim n→∞ z j n (x 0 ) = 0. Consequently, by (12) and (13), Note that for p = N the inequality |s N | 2 < |s p | always holds. Hence we get We have the following criterion of the convergence of the sequence {S −n F n (x)}.
Proof. Fix j < p and define By (12) we have Since |s i1 . . . s i k | < |s j | for (i 1 , . . . , i k ) ∈ A j and the sequences {y i n (x)} are bounded, the sequence {z j n (x)} converges uniformly in V to zero and, consequently, by (13) and (12), the j-th coordinate of {S −n F n (x)} converges to the j-th coordinate of the Schröder function. For the coordinates j ≥ p the assertion follows from Theorem 4.
In a particular case for N = 2 and r = 3 Theorem 5 has the following simple form ∂x2∂x2 = 0 and the sequence S −n F n (x) is bounded, then formula (2) holds.
where g is smooth, g(0, 0) = 0, dg(0, 0) = 0, d (2) g(0, 0) = 0 and g is chosen such that the sequence {S −n F n (x, y)} is bounded. It is easy to verify that the formal solution η of the second order is given by the formula η(x, y) = (x + xy, y) and satisfies the assumptions of Corollary 2. In the particular case when g(x, y) = 0 it is easy to calculate that , y .
In this case the above sequence converges to (x + xy, y).
Proof. If k ≥ 3, then |s i1 . . . s i k | ≤ |s N | k < |s N | 2 . Let k = 2 and suppose that there existī 1 ,ī 2 such that |sī 1 sī 2 | = |s N | 2 . Then |s N | 2 ≤ |sī 1 ||s N |, hence |s N | ≤ |sī 1 |, so |sī 1 | = |s N |, and consequently also |sī 2 | = |s N |. Since Lemma 4. If c n := a 1 n (q 1 ) n + · · · + a k n (q k ) n + b n q n , |q i | < |q|, i = 1, . . . , k, |q| > 1 and the sequences {c n } and {a i n } are bounded, then lim n →∞ b n = 0. Proof. The inequality |c n | < M implies that | cn q n | ≤ M q n → 0, so By the boundedness of {a i n } we get the thesis. Lemma 5. Let r ≥ 3, hypotheses (i) and (H r ) be satisfied and the matrix S be diagonal. If for every x in a neighbourhood of the origin the sequence {S −n F n (x)} has a bounded subsequence and |s N −1 | < |s N | and |s p | < |s N | 2 then a 1 NN = · · · = a p N,N = 0, where a j k,l are the coefficients defined by (11). Proof. Let ϕ be a C r Schröder function of F . By Lemma 2 there exists a point x N such that ϕ N (x N ) = 0. Let {n ν } be a sequence such that the subsequence {S −nν F nν (x N )} is bounded. Let j ≤ p. By (13) and (14) we infer that the sequence {z j nν (x N )} is bounded. The expression (12) may be written in the following form Since |s i | ≤ |s N | < 1 we have qî := |s −j s i1 . . . s i k | < |s −j s 2 N | =: q for k ≥ 3, however, for k = 2 the above inequality is true except for the case i 1 = i 2 = N . Moreover, On the other hand, by Remark 3, lim n→∞ y N nν (x N ) = ϕ N (x N ) = 0, so a j N,N = 0, for j ≤ p.
From the argumentation of the above proof and Remark 3 we get the following The following result shows that for r ≥ 3 formula (6) in Theorem 1 cannot be replaced by (2). Theorem 6. Let r ≥ 3 and hypotheses (i), (H r ) be satisfied, S be diagonal, |s N | r < |s 1 |, |s N −1 | < |s N | and η = (η 1 , . . . , η N ) be the formal solution of (1) of order 2. If ∂ 2 η p ∂xN ∂xN (0) = 0 for a p such that |s p | < |s N | 2 , then in every neighbourhood of the origin there exists x 0 = 0 such that the sequence Proof. The assertion is a consequence of Lemma 5, Corollary 3 and the equalities d (2) ϕ(0) = d (2) η(0), This theorem gives the answer to the main question of the paper. More precisely we have Corollary 4. Let r ≥ 3 and F satisfy the assumptions of Theorem 3. Let dF (0) be a diagonal matrix and |s N −1 | < |s N |. If ϕ is a Schröder function of F and ∂ 2 ϕ p (0) ∂xN ∂xN = 0 for a p such that |s p | < |s N | 2 , then formula (2) does not hold.
We give an example of a function F satisfying the assumptions of Theorem 6 and Corollary 4.
Proof. We have . . . It follows, by Proposition 1 in the one dimensional case, that there exists a δ > 0 such that for every 1 ≤ i ≤ N there exists the limit lim n→∞ F n i (u) s n i = ϕ i (u) for |u| < δ and that ϕ i are of class C r . Hence we get (2).
Remark. Note that if the sequence {S −n F n (x)} is convergent to ϕ(x), then F n (x) ≈ S n ϕ(x), which means that the sequence {F n (x)} has the same asymptotic behavior as the sequence S n ϕ(x). The matrices sequences S n are well described in the literature (see, for e.g., Elaydi [4]). As an application of this fact we obtain the assertion that the solutions of the system of nonlinear difference equationŝ x n+1 = F (x n ) in R N space can be approximated by the solutions of a linear systemŷ n+1 = Sŷ n with the initial conditionŷ 0 = ϕ(x 0 ).