Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators Cφ:AR→AR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}$$\end{document} for a real analytic self map φ:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi:\mathbb{R} \rightarrow \mathbb{R} }$$\end{document} as well as an isomorphic description of corresponding eigenspaces. We completely characterize those φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document} for which Abel’s equation f∘φ=f+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f\circ \varphi=f+1}$$\end{document} has a real analytic solution on the real line. We find cases when the operator Cφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_{\varphi}}$$\end{document} has roots using a constructed embedding of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document} into the so-called real analytic iteration semigroups.


Introduction and Preliminaries
Let ϕ : R → R be a non-constant real analytic map and let A (R) be the space of real analytic functions defined on R. Each symbol ϕ : R → R defines a composition operator C ϕ : When A (R) is endowed with its natural locally convex topology (see below), C ϕ is a continuous linear operator on A (R). The first purpose of this article is to determine the eigenvalues and eigenvectors of composition operators C ϕ : A (R) → A (R).
Of course, this is just to find a solution f of the equation This is a very classical topic. The equation appeared probably for the first time already in 1871 in a paper of Schröder [38] and was partially solved in 1884 in a paper of Königs [33] also for real analytic functions. That is why (1) is often called the Schröder equation. There is an impressive bibliography of the subject (see [34,Chapter 6], [35,Chapter 9] and citations therein or a J. Bonet and P. Domański IEOT survey paper [8,Sec. 8], as well papers on the holomorphic case [18,40,41]) but, in spite of extensive literature inquires, we could not find any known complete solution of the problem in our setting. The first main result of the paper provides such a full solution describing also corresponding eigenspaces (later on we also describe the eigenfunctions). In the rest of the article we denote id (x) = x, x ∈ R, and, for a map ϕ : R → R, we write ϕ [0] = id and ϕ [n] for the n-times composition of ϕ, n ∈ N.
Theorem A. Let ϕ : R → R be a real analytic map. Then the map C ϕ : A (R) → A (R) has the following set of eigenvalues: (a) C \ {0} whenever either ϕ > id and the set of critical points of ϕ is bounded from above or ϕ < id and the set of critical points is bounded from below-in this case every eigenspace is topologically isomorphic to the space A (T) of real analytic functions on the unit circle T; (b) {(ϕ (u)) n : n = 0, 1, 2, . . . } whenever ϕ [2] has exactly one fixed point u and one of the following cases hold: • 0 < |ϕ (u)| < 1, • 1 < |ϕ (u)| and ϕ has no critical points -in this case eigenspaces are one-dimensional of the form span{f n } where f is an eigenvector of C ϕ with the eigenvalue ϕ (u); (c) {1, −1} whenever ϕ [2] = id = ϕ-in this case the whole space is a direct sum of eigenspaces and each of them is isomorphic to the space A + (R) of even real analytic functions on R; The fact that for ϕ with a fixed point u the eigenvalues λ are powers of ϕ (u) was already known to Königs (comp. [32,Satz 3]) who also proved local existence in the case (b). Since under the assumption of (b) in that case the attraction basin of the fixed point u is the whole R, then this particular case above is known (see for instance [34,Th. 6.4,6.5], comp. [32,Satz 1]). The case (c) is also known see [34,Th. 6.10]. Part (a) is closely related to the so-called Abel equation ( In fact, any solution f to the Abel equation produces a solution g, g(x) := exp(log(λ)f (x)) of the Schröder equation for any λ = 0 see [32, p. 57]. The part (a) of Theorem A above is a consequence of our second main result characterizing when the Abel equation is solvable in A (R) (in view of the above remarks this case contains the main novelty in Theorem A). Note that the isomorphic classification of eigenspaces as well as the last sentence of Theorem A seems to be also new. has a real analytic solution f : R → C if and only if ϕ has no fixed points and the set of critical points of ϕ is bounded from above (in case ϕ > id ) or from below (in case ϕ < id ). Moreover, in that case there is a real analytic solution f 0 : R → R such that its set of critical points is bounded from above (in case ϕ > id ) or from below (in case ϕ < id ) and for every such f 0 each solution f of the Abel equation is of the form f (x) = f 0 (x) + g • f 0 (x), where g : R → R is an arbitrary 1-periodic real analytic function.
The Abel equation is another classical subject. It was probably mentioned for the first time by Abel [1] in his note published posthumously (comp. also [45]) when a relation between f and f 0 as above was given in the case of strictly increasing solutions of the Abel equation with some additional assumptions on g. There is also a broad literature about the equation in various function classes [34,Chapter 7] or [8,Sec. 9]. There are also recent papers [45,46], and papers on the holomorphic case, see for instance [15][16][17][18]. So far the Abel equation was solved in real analytic functions globally on R for ϕ = exp by Kneser (see [32, p. 64], comp. also a series of papers of Walker [48][49][50][51] for ϕ(x) = exp(x) − 1 or ϕ(x) = exp(bx) inspired partially by numerical analysis) and there was a characterization of real analytic diffeomorphisms ϕ for which the Abel equation is solvable (iff ϕ has no fixed point, see [10,Main Theorem], comp. [11,Th. 3.6]). In [9,Th. 1.4,Cor. 4.2] it was proved that a necessary condition for real analytic solvability of the Abel equation is that all compact sets K ⊂ R are wandering, i.e., that there is ν ∈ N such that for n, m ∈ N, |n − m| > ν holds ϕ [n] This condition is strictly weaker than the condition we found.
The motivation of Kneser [32] for solving the Abel equation comes from his problem of finding an iteration root of exp, i.e., of a real analytic function r such that r [2] = exp. It is a folklore that if f is an invertible solution of the Abel or Schröder equations (λ > 0) then , respectively, is a so-called real analytic iteration semigroup in which ϕ embeds, i.e., G : (R + ∪ {0}) × R → R is real analytic satisfying the following conditions is the required root. We prove the third main result of the paper: Theorem C. A real analytic map ϕ : R → R embeds into a real analytic iteration semigroup whenever ϕ has no critical points and either ϕ has no fixed points or ϕ [2] has exactly one fixed point u and 0 < ϕ (u) = 1. In particular in that case there exist roots of the operator C ϕ of arbitrary order.
The composition operator is definitely one of the most natural linear operators of analysis and there is an extensive literature on that subject: see the monographs in case of spaces of holomorphic functions [19,39] and the papers on a real analytic case [13,14,[20][21][22][23][24]. For a literature on the space of real analytic functions see a recent survey [20]. The paper is closely related to questions of real analytic (nonlinear) dynamics of one variable see [11] which in turn is connected with its holomorphic counterpart [37]. For the theory of one-dimensional manifolds we refer to [47] although the results are very classical and known much earlier. For functional analytic tools see [36]. Now, we present the organization of the paper in which we look at the considered problems from the functional analytic or operator theoretical point of view. We discuss in Sect. 2 the case when the self map ϕ has fixed points. Our main result is Theorem 2.9, that gives a complete picture of the eigenvalues, eigenvectors and eigenspaces of the composition operator C ϕ : A (R) → A (R) when ϕ has fixed points. Section 3 studies the case when ϕ has no fixed points and Theorem 3.11 gives several characterizations of those self maps ϕ : R → R such that the Abel equation (or a specific Schröder equation) has a real analytic solution. The method used is inspired by the method used in [10] (comp. [9]) via the orbit space R/ϕ of the real analytic map ϕ. Our approach allows us to give in Sect. 4. Theorem 4.2 a characterization of fixed point free ϕ which embeds into a real analytic iteration semigroup and give some sufficient conditions for in the case of ϕ with fixed points (Proposition 4.3). This implies an extension of Kneser's [32] and Belitskii and Tkachenko's [11,Th. 2.20] results about the existence of real analytic functions g such that g • g = ϕ.
In what follow N 0 denotes the set of natural numbers N and 0. The open interval in R with extreme points a < b is denoted by (a, b). Recall that we denote by id : R → R the identity map id (x) = x and by I : A (R) → A (R) the identity operator on A (R). The ball in the complex plane of center z and radius r > 0 is denoted by B(z, r). If T is a continuous linear operator on a locally convex space E, its kernel and image are denoted respectively by ker T and im T . The point spectrum σ p (T ) of T is the set of all λ ∈ C such that T − λI is not injective. Elements of σ p (T ) are called eigenvalues of T . The spectrum σ(T ) of T is the set of all λ ∈ C such that T − λI is not a topological isomorphism from E onto E. By the open mapping theorem which works for surjective endomorphisms of A (R) (see [20] A description of the natural topology on A (R), that goes back to Martineau, is given, for instance, in [25]. The space A (R) has very good properties: it is nuclear, separable, complete, satisfies the closed graph theorem and the uniform boundedness principle, but surprisingly it has no Schauder basis by [25]. To be precise, the space A (R) is equipped with the unique locally convex topology such that for any where (U n ) n∈N is a basis of C d -neighbourhoods of K. Martineau proved that there is exactly one topology on A (R) satisfying the condition above. For our purposes, it is important to recall that a sequence (f n ) n∈N in A (R) tends to f if and only if there is a complex neighbourhood W of R such that each f n and f extend to W holomorphically and f n → f uniformly on compact subsets of W . The topology of A(J) for an open interval J in R is defined analogously. A long survey on spaces of real analytic functions with a very precise description of their topology is contained [20].
The following easy result is included for references in the rest of the paper. (3) Let ϕ : R → R be a non-constant analytic map. If ϕ is a real analytic diffeomorphism, it is easy to see that C ϕ is bijective. If C ϕ is surjective, there is f ∈ A (R) such that f (ϕ(x)) = x for all x ∈ R. This implies right away that ϕ is injective in R. As it is continuous, the image ϕ(R) is an interval J and ϕ is strictly increasing or decreasing with a continuous inverse ϕ −1 . Clearly f (y) = ϕ −1 (y) for each y ∈ J. This implies that ϕ −1 is real analytic in J and it is the restriction to J of the real analytic f defined on the whole R. If J is not the whole real line, let a be an extreme of the interval. By continuity, ϕ −1 (y) tends to +∞ or −∞, and ϕ −1 cannot be the restriction of f ∈ A (R) to J. Summarizing, ϕ is bijective from R onto itself, and the inverse is real analytic. Now (4) is a direct consequence of (3).
We conclude this section recalling when composition operators on A (R) are open, see [22].
Proof. (a) If ϕ has no critical points, then it is a diffeomorphism onto im ϕ = (a, b), where a could be −∞ and b could be +∞. Hence C ϕ is an isomorphism If ϕ is surjective, then it is semi-proper, i.e., for any compact set K ⊂ R there is a compact set L ⊂ R such that ϕ(L) = K. Then the result follows from [22,

Self Map with Fixed Points
First, we describe precisely eigenvalues and eigenvectors in the case when ϕ has a fixed point. As it was explained in the introduction much about this case is known. Nevertheless we present the results and some proofs for the convenience of the reader, since the results are scattered in the literature. A description when eigenspaces span densely A (R) and the isomorphic classification of the eigenspaces seems to be new.
Proof. We distinguish several cases depending on the value of ϕ (u). Case 1. If |ϕ (u)| < 1, then the fixed point u is attractive and we can find a sequence (x n ) n of pairwise distinct points converging to u and such that ϕ(x n ) = x n+1 . Thus (b) in the statement is satisfied. Case 2. If |ϕ (u)| > 1, then the fixed point u is repelling and there is ε such that |ϕ (y)| > 1 for each y ∈ (u − ε, u + ε). This implies that either ) and define (x n ) n the sequence of the iterates of x 1 by the inverse ϕ −1 . It is easy to see that the iteration can be accomplished and that (x n ) is a sequence of pairwise different points converging to u for which condition (b) in the statement holds.
Case 3. If ϕ (u) = 1, there are two possible subcases. The first one is that ϕ (x) = 1 for each x ∈ R. In this case ϕ(x) = x for each x ∈ R. On the other hand, if we are not in this case, there is a neighbourhood of u in which ϕ does not coincide with 1. Otherwise, since ϕ is real analytic, it would follow ϕ(x) = x for each x ∈ R. Now we have either ϕ (y) > 1 for each y ∈ (u, u + ε) or 0 < ϕ (y) < 1 for each y ∈ (u, u + ε) for some ε > 0, and similarly at the other side (u − ε, u). Proceeding as in cases 1 and 2 in the side where we have the inequality, we can find the desired sequence (x n ) for which statement (b) holds.
Let us observe that the map f → (x → xf (x)) is an isomorphism of the space A + (R) of even real analytic functions onto the space A − (R) of odd real analytic functions. Proposition 2.2. Let ϕ : R → R be a real analytic map with a fixed point u ∈ R. The following holds: (ii) If ϕ [2] = id but ϕ = id , then C ϕ has only two eigenvalues 1 and −1 and ϕ (u) = −1. In this case each f ∈ A (R) can be decomposed as is an eigenvalue with eigenvector 1 (resp. −1). In this case both eigenspaces are isomorphic to the space of even real analytic functions It is clear that the decomposition is unique and using this decomposition one proves that no λ ∈ C \ {1, −1} is an eigenvalue.
In order to prove that ϕ (u) = −1 in part (ii), we assume without loss of generality that u = 0. We observe that the graph of ϕ has to be symmetric with respect to the line y = x. Moreover, differentiating the equation ϕ(ϕ(x)) = x we get ϕ (u) · ϕ (u) = 1, so ϕ (u) = 1 or ϕ (u) = −1. In the first case, since ϕ is not the identity map but increasing around u, there is a neighbourhood of u where ϕ(x) never takes value x except for u. For instance, for some ε > 0 and every x ∈ (−ε, 0) we have x < ϕ(x) < 0. Such a function cannot be symmetric with respect to the line y = x. A similar proof works for the case ϕ(x) < x. We have proved that ϕ (u) = −1.
Since ϕ [2] = id , ϕ is a real analytic diffeomorphism of R onto R so it has no critical points. Thus for every x ∈ R we have ϕ (x) < 0, in particular, ϕ has the only one fixed point u and lim x→+∞ ϕ(x) = −∞ and lim x→−∞ ϕ(x) = +∞. Define The map ψ is strictly increasing, it has no critical points and it is surjective since Hence C ψ is an isomorphism of A (R). Let f ∈ A (R) be even then Analogously for odd f we have C ψ (f ) • ϕ = −C ψ (f ). We have proved that C ψ is an isomorphism of the space of even/odd real analytic functions IEOT If the assumption of (iii) holds, we can apply Lemma 2.1 to find a convergent sequence (x n ) n in R of such that for each n = k we have x n = x k and there is m such that ϕ [m] Any other real analytic function g such that C ϕ (g) = g would be constant on the convergent sequence of pairwise different points (x n ) n . This implies that g is constant.
Proceeding by contradiction, if the conclusion does not hold, we apply Lemma 2.1 to find a convergent sequence of pair- Passing to a subsequence, it follows that the real analytic function f is constant. Proof. If ϕ [2] = id , then ϕ has more than one fixed point if and only if ϕ = id .
Theorem 2.6. Let ϕ : R → R be a real analytic function with a fixed point u such that 0 < |ϕ (u)| < 1. Then either (a) ϕ [2] has at least two fixed points and then 1 is the only eigenvalue of C ϕ ; (b) or (ϕ (u)) n is an eigenvalue for every n ∈ N. Proof. Define Ω to be the basin of attraction of u, i.e., and the open interval joining u = ϕ(u) and ϕ(z) contains only points of Ω. We have proved that if Ω 0 is the connected component of Ω containing u then ϕ(∂Ω 0 ) ⊂ ∂Ω 0 . It is easily seen that at least one point in ∂Ω 0 is fixed for either ϕ or ϕ [2] . Summarizing, we have shown that either ϕ [2] has at least two fixed points or the whole real line is the basin of attraction for u. Thus the statement (a) follows from Propositions 2.5. (comp. the proof of Proposition 2.7 (a) below). Now, we consider the second case. It is easy to see that if (ϕ (u)) n is an eigenvalue of C ϕ with an eigenvector f , then (ϕ (u)) nk is an eigenvalue with an eigenvector f k . Accordingly, it suffices to prove the result for n = 1. The construction is now like [ [34,Th. 6.5]). The statement (b) is thus known (comp. Proposition 2.4). However, the part about the form of the closed linear span of eigenspaces seems to be new and we prove it below. Observe that in [12,Th. 4.5] it is proved that if ϕ is a diffeomorphism with several fixed points on which |ϕ | = 1 then all eigenspaces are finite dimensional (even for more general operators).
Since ϕ (u) is real the real part of every eigenvector is also an eigenvector. It is clear that the closed linear span of all the eigenspaces is equal to lin {f n : n ∈ N} for any eigenvector f of C ϕ corresponding to the eigenvalue ϕ (u). Since polynomials are dense in A (R) we have for real valued f : Proposition 2.7. Let ϕ : R → R be a real analytic function with a fixed point u such that 1 < |ϕ (u)|. The following holds (a) If ϕ [2] has at least two fixed points, then 1 is the only eigenvalue of C ϕ ; (b) ϕ (x) = 0 for some x ∈ R and ϕ [2] has only one fixed point, then 1 is the only eigenvalue of C ϕ (c) If ϕ (x) = 0 for each x ∈ R and ϕ [2] has only one fixed point, then (ϕ (u)) n is an eigenvalue for every n ∈ N with a real valued eigenfunction. Moreover, in that case, the closed linear span of all eigenspaces is equal to A (R) and every real valued eigenfunction f for an eigenvalue ϕ (u) is a diffeomorphism of R onto R.
Vol. 81 (2015) Abel's Functional Equation on Spaces 465 We are ready for the proof of (b). Suppose that ϕ satisfies ϕ (x 0 ) = 0 for some x 0 ∈ R and that ϕ [2] has only one fixed point. Then ϕ [2] satisfies the assumptions of the proof just given, but ϕ [2] = id . So 1 is the only eigenvalue of C ϕ [2] and, by Proposition 2.3, −1 is not the eigenvalue of C ϕ . So the proof of part (b) is complete.
Let f be an eigenvector of C ϕ for the eigenvalue ϕ (u) and of C ϕ −1 for the eigenvalue (ϕ (u)) −1 . By  We may take f real valued by taking the real part of any eigenfunction. Since f (u) = 0 and f (u) = 0, f takes both positive and negative values. Since |ϕ (u)| > 1 and it follows that f is surjective on R. We have proved that f : R → R is a diffeomorphism.
Finally, the eigenspace for (ϕ (u)) n is equal to lin {f n }. The linear span of all eigenspaces is equal to the image of the subspace of polynomials in A (R) by the topological isomorphism C f : A (R) → A (R) so dense in A (R) (see Theorem 1.2).
Let us note that from Theorem 2.6 and Proposition 2.7 the values (ϕ (u)) n are sometimes eigenvalues and sometimes they are not. Surprisingly, they are always elements of the spectrum by a result that is proved with a technique due to Hammond [28,Prop. 4.1].

Proposition 2.8. Let ϕ : R → R be a real analytic function and let C ϕ :
A (R) → A (R) be the associated composition operator. If u is a fixed point of ϕ such that |ϕ (u)| = 1, 0, then ϕ (u) n ∈ σ(C ϕ ) for each n ∈ N 0 .
Proof. If n = 0, then ϕ (u) n = 1 ∈ σ p (C ϕ ) by Proposition 1.1. Fix n ∈ N. Proceeding by contradiction, assume that there is f ∈ A (R) such that Since |ϕ (u)| = 1, f (u) = 0. Suppose by induction that f (k) (u) = 0, 0 ≤ k ≤ j − 1. Taking the j-th derivative in the equality above, j < n, we get The first term consists of ϕ (u) j f (j) (u) plus some other summands involving lower order derivatives of f at u, that vanish by the induction hypothesis. Therefore hence f (j) (u) = 0. Now taking the n-th derivative we reach a contradiction: Summarizing, we have obtained complete description of eigenvalues and the dimension of the corresponding eigenspaces for all C ϕ whenever ϕ has a fixed point as follows (so it completes the proof of Theorem A in the fixed point case). (i) either ϕ [2] = id but ϕ = id and in this case there are two eigenvalues ±1 and A (R) is a direct sum of two eigenspaces (ii) or ϕ [2] = id , 1 is the only eigenvalue and its eigenspace is onedimensional. (c) If ϕ (u) = 0, then 1 is the only eigenvalue and its eigenspace is onedimensional. (d) If 0 < |ϕ (u)| < 1, then (i) either ϕ [2] has at least two fixed points and then 1 is the only eigenvalue and its eigenspace is one-dimensional (ii) or ((ϕ (u)) n ) n∈N is the sequence of eigenvalues and all of them have one-dimensional eigenspaces. (e) If 1 < |ϕ (u)|, then (i) either ϕ [2] has at least two fixed points or ϕ has a critical point and then in both cases 1 is the only eigenvalue and its eigenspace is one-dimensional (ii) or ((ϕ (u)) n ) n∈N is the sequence of eigenvalues and all of them have one-dimensional eigenspaces.
Our results in this section permit us to determine the eigenvalues and the eigenspaces of

Self Map Without Fixed Points and the Abel Equation
Let ϕ : R → R be a real analytic function without a fixed point. Recall that the Abel equation is the equation Clearly Proof. Observe first that the function f 0 cannot be constant. Let p be a periodic function with period 1 and define f : Thus C ϕ (f ) = f . The infinite dimensionality follows varying p. This settles the case λ = 1. Taking a non-vanishing periodic function p we get the desired non vanishing eigenvector.
Hence G is an eigenvector of C ϕ with an eigenvalue λ.
If F ∈ A (R) is a fixed point of C ϕ (and there is an infinite dimensional subspace of such functions), we get C ϕ (F G) = λF G. This implies that the eigenspace of the eigenvector λ is also infinite dimensional.
Let y ≥ a > m ϕ , then we construct a decreasing sequence ψ [n] (y) for all n such that ψ [n−1] (y) > m ϕ . If this sequence is infinite and bounded from below by a, it has a limit and then the set of critical points of ϕ is bounded from above; in particular, ϕ is strictly increasing from some point on.
Remark 3.6. Of course, an analogous result holds for ϕ < id if the set of critical points of ϕ is bounded from below.
Proof. We start with some claims. Claim 1. If ϕ [n] has a critical point of order k at x 0 (i.e., the derivative has a zero of order k at x 0 ), then f has a critical point of order k at x 0 .
Proof of the Claim 1. Since . It is easily seen that for l = 1, . . . , k This completes the proof of Claim 1.
Claim 2. If for some x 0 there is a sequence 0 ≤ n 1 < n 2 < · · · < n k of natural numbers such that ϕ [nj ] (x 0 ) for j = 1, . . . , k are critical points of ϕ, then ϕ [n] for n > n k has a critical point of order ≥ 2 k − 1 at x 0 .
Proof of Claim 2. If ϕ has a critical point of order l at x and of order m at ϕ(x), then ϕ [2] has a critical point of order ml + m + l at x. Indeed, let us take Hence ϕ [2] Thus x is a critical point of order (ml + m + l + 1) − 1. Inductively we will get Claim 2. Now, we prove our Lemma. Let a > m ϕ . Then by Lemma 3.4, for every y ≥ a there isŷ ∈ [a,φ(a)] such that y = ϕ [n] (ŷ) for some n ∈ N. IEOT If ϕ has infinitely many distinct critical points (x n ) n∈N ⊂ [a, ∞), then there is a sequence (x n ) n∈N ⊂ [a,φ(a)] such that Case (a). There are infinitely many distinct points in the sequence (x n ) n∈N . By Claim 1, there are infinitely many distinct critical points of f on [a,φ(a)]. Thus the set of critical points has an accumulation point and thus f ≡ 0; a contradiction.
Case (b). There is a pointx ∈ [a,φ(a)] and an infinite increasing sequence (k n ) such that for all n ∈ N the number ϕ [kn] (x) is a critical point of ϕ. By Claim 2, for any m > k n the numberx is a critical point of ϕ [m] of order ≥ 2 n − 1. By Claim 1, f has a critical pointx of order ∞ so f ≡ 0; a contradiction. Now, we are ready to formulate our necessary condition for real analytic solvability of the Abel equation (the condition turns out to be sufficient as well). So far the best necessary condition for solvability of the Abel equation is due to Belitskii and Lyubich [9, Th. 1.4]: all compact sets in R must be wandering, i.e., for every compact K there is an integer ν such that for any two n, m ∈ N such that |n−m| > ν the sets ϕ [n] (K) and ϕ [m] (K) are disjoint. This condition is strictly weaker than our condition. For diffeomorphisms ϕ the condition is equivalent with lack of fixed points but in general it only implies that ϕ is fixed point free. It is known that for solvability in continuous functions of all equations f • ϕ = f + γ for every continuous function γ it is necessary and sufficient that ϕ has no fixed points and ϕ is strictly increasing on some set [c, +∞) (in case ϕ > id ) or on (−∞, c] (in case ϕ < id ), see  Proof. We consider only the case ϕ > id -the other one is analogous. If λ = 1 then the eigenvectors cannot be constant. The result follows from Lemma 3.5.
Next part is devoted to the proof of sufficiency of our condition. The method of the proof is "topological" in nature and inspired by [10]. In that paper only the case of diffeomorphisms ϕ is considered. Transferring it to the general real analytic functions ϕ required some new ideas.
Let ϕ : R → R be a real analytic map. Then for any x ∈ R we denote by O(x) the full orbit of x via ϕ, i.e., Vol. 81 (2015) Abel's Functional Equation on Spaces 471 The full orbits form a partition of R. The quotient topological space with respect to that partition is denoted by R/ϕ and the corresponding (continuous) canonical quotient map we denote by π ϕ : R → R/ϕ. It is worth noting that R/ϕ need not be Hausdorff. By Lemma 3.4 we get: Corollary 3.8. Let ϕ : R → R be a real analytic map without fixed points. Then R/ϕ is compact.
Proof. We consider only the case ϕ > id . Then for any w ∈ R its full orbit O(w) is unbounded from above. By Lemma 3.4, there is z ∈ O(w) ∩ [a,φ(a)] for any fixed a > m ϕ . We have proved that π ϕ maps continuously [a,φ(a)] onto R/ϕ. This completes the proof by [26, 1.1.7.8].
Now, we study the natural manifold structure on R/ϕ. The next two results are generalizations to the case of non-diffeomorphic ϕ of the method presented by Belitskii and Lyubich in [10, Th. 3.1]. Lemma 3.9. Let ϕ : R → R be real analytic and ϕ > id . If the set of critical points of ϕ is bounded from above, then R/ϕ is homeomorphic to the circle T and there is a real analytic structure on R/ϕ which makes it diffeomorphic to T and makes the canonical map π ϕ : R → R/ϕ real analytic, such that its set of critical points coincides with the set of critical points of all the maps ϕ [n] for n ∈ N.
Proof. Assume that every critical point of ϕ is strictly smaller than a, in particular ϕ is strictly increasing on [a, +∞). Hence for x ≥ a we have ϕ(x) = ϕ(x). By Lemma 3.4, [x, ϕ(x)] intersects every full orbit. If a ≤ x < y < z < ϕ(x) then x < ϕ [n] is an open set in R. Therefore π ϕ ((y, z)) is an open set.
We have proved that for every x ≥ a the map π ϕ restricted to (x, ϕ(x)) is a homeomorphism onto an open set (clearly a Hausdorff one). Since every pair of distinct full orbits O(y), O(z) is contained in one of these open sets it follows that the whole space R/ϕ is Hausdorff. Summarizing, by Corollary 3.8, the space R/ϕ is a connected compact Hausdorff one dimensional locally euclidean space, i.e., a connected compact manifold of dimension 1 without boundary. By [47,Th. 3.2], R/ϕ is homeomorphic to the circle T. Now, we define a real analytic structure on R/ϕ. For any x ≥ a we define a chart . We consider π ϕ ((x, ϕ(x)) ∩ π ϕ ((y, ϕ(y)) then Therefore, , Since for every m ∈ N the map ϕ [m] has no critical points on [a, +∞) the above maps are real analytic.
We have proved that the charts defined above form a real analytic atlas. Moreover, the real analytic structure on R/ϕ must be diffeomorphic to the standard one on T by [47,Th. 6.3].
Let us take any point y ∈ R then there is n ∈ N such that ϕ [n] (y) > a so some neighbourhood U of y is mapped via ϕ [n] into a neighbourhood of ϕ [n] (y), contained in (x, ϕ(x)) where x > a. Now, the map is real analytic on U . The map π ϕ has a critical point in y if and only if ϕ [n] has it.
We will make use of the notion of the contractible map by which we mean a map homotopic to a constant one. Proof. Let us denote by q : R → T, q(x) = exp(2πix), the standard quotient map. Clearly, q is a local diffeomorphism. Since R is contractible then, by [27,Th. 6.1] the map d • π ϕ : R → T lifts to a continuous map Φ : R → R with respect to q, i.e., the following diagram commutes: Vol. 81 (2015) Abel's Functional Equation on Spaces 473 Since q is a real analytic local diffeomorphism then the lifting Φ has to be real analytic and its critical points are exactly critical points of d • π ϕ . Moreover, Φ(ϕ(x)) = Φ(x) + n(x), where n : R → Z is a continuous (hence constant) function n(x) := n. Now, we will use the well-known facts concerning liftings of homotopy but for the reader's convenience we provide details. If n = 0 then we can define a continuous map and F (·, t) is constant on full orbits of ϕ for every t ∈ [0, 1]. We can define a continuous map and q • G is a homotopy joining d with the constant function; a contradiction.
We have proved that n = 0 so we can define f :  case ϕ > id ) or bounded from below (in case ϕ < id ). In that case for every complex λ = 0, e μ = λ, the map is a topological isomorphism of A (T) onto the eigenspace of C ϕ for λ (here q : R → T, q(x) := exp(2πix)).
. This result improves the result of Abel true only for strictly increasing solutions; see [1] or [45]. We prove now (b ), (c)⇒(d): first of all, by Theorem 2.9, (c) implies that ϕ has no fixed point, and (b ) implies that there is a non-constant eigenvector for some complex eigenvalue λ = 1 (comp. Proposition 1.1). Since both cases ϕ > id and ϕ < id are analogous, we assume ϕ > id , and the conclusion (d) follows from Lemma 3.5.
(d)⇒(e) is Lemma 3.9 in case ϕ > id , the other case is analogous. (e)⇒(f): by [47,Th. 6.3], R/ϕ is real analytic diffeomorphic to T and, obviously, this diffeomorphism is a non-contractible real analytic map d from R/ϕ to T. Apply Lemma 3.10 to conclude.
So far we have completed the proof of the equivalence of all the assertions.
For a real eigenvalue λ with a non-vanishing eigenvector f , the real part Re f and the imaginary part Im f of f are also such vectors. Clearly, (Re f ) 2 + (Im f ) 2 is a strictly positive eigenvector for the eigenvalue λ 2 .
It remains to show the last part of the statement. We consider only the case ϕ > id . By (e), there is a diffeomorphismd : R/ϕ → T and as in the proof of Lemma 3.10 we produce a solution of the Abel equation f 0 : R → R. By Lemmas 3.10 and 3.9, f 0 has critical points exactly at critical points of Vol. 81 (2015) Abel's Functional Equation on Spaces 475 ϕ [n] for any n ∈ N so its set is bounded from above. Since exp •(μf 0 ) is a non-vanishing function and we get that the map is a topological isomorphism. So it suffices to consider the case λ = 1, μ = 0 and T 1 = C q•f0 . It is easy to observe that T 1 : A (T) → ker(C ϕ − I) continuously. Since f 0 (R) is a halfline unbounded from above (or the whole line) we get q•f 0 (R) = T. Then by [21,Th. 3.2], C q•f0 is open onto its image. Since C q•f0 is injective it follows that this map is a topological isomorphism. Now, we can prove that C q•f0 is surjective onto ker(C ϕ − I) for the above choice of f 0 . Let C ϕ (f ) = f , then f is constant on every full orbit O(x), x ∈ R. Thus f =f • π ϕ for some continuous mapf : R/ϕ → C. Since π ϕ is a local diffeomorphism on a halfline of R mapped via π ϕ onto the whole R/ϕ the mapf is also real analytic.
Let us take the map q • f 0 : R → T, clearly q • f 0 is constant on full orbits of ϕ so q • f 0 = d • π ϕ for some continuous map d : R/ϕ → T. Since q • f 0 has no critical points on a halfline of R unbounded from above which is mapped via π ϕ onto the whole R/ϕ, π ϕ is there a local diffeomorphism and q • f 0 (R) = T then d : R/ϕ → T is real analytic, surjective and has no critical points. We will show that d is injective. In order to show that we have to show that q • f 0 does not glue together two different orbits. Now, for x big enough ϕ is strictly increasing on (x, ϕ(x)) and also f 0 is injective on the same interval. Moreover, f 0 (ϕ(x)) = f 0 (x) + 1. This implies that for any y ∈ (x, ϕ(x)) the difference between f 0 (x) and f 0 (y) is non-integer. On the other hand by Lemma 3.4 every orbit different from O(x) intersects (x, ϕ(x)). We have proved that q•f 0 cannot glue this orbit with O(x). Finally, d : R/ϕ → T is a diffeomorphism. Therefore Theorem 3.11 can be used to conclude that the Abel equation f (ϕ(x)) = f (x) + 1 has a real analytic solution f ∈ A (R) for ϕ(x) = e αx , α > 1/e (for α ≤ 1/e the map ϕ has a fixed point), and it does not have a real analytic solution for ϕ(x) = x + 1 + α sin(α −1 x), 0 < α < 1, since this function is real analytic, it is a (continuous) homeomorphism on R, has no fixed points, but it has an unbounded sequence of critical points, namely ϕ (x) = 0 if and only if x = 2sπα, s ∈ Z. The later example is mentioned in [9, Example 6.2] in connection with the smooth solvability of the cohomological equation. Observe that the point spectrum of C ϕ for ϕ(x) = e αx , α > e −1 is equal to C \ {0} (the spectrum is equal to C), and that for ϕ(x) = x + 1 + α sin(α −1 x), 0 < α < 1, the only eigenvalue of C ϕ is 1. Kneser [32] considered the case ϕ = exp, Belitskii and Tkachenko [11,Th. 2.20]proved that (c) implies existence of arbitrary iteration roots without using iteration semigroup. They also showed in the same place that if ϕ is a diffeomorphism, then (a) holds.
(b)⇒(a) Solvability of the Abel equation implies that ϕ has no fixed point. Since f has no critical points it is either strictly increasing or strictly decreasing. The image of f is always unbounded from above so it must be of the form (a, +∞) where a could be −∞. It is easily seen (in fact this is an observation of Abel [1], comp. [34, p. 198] and [32, p. 57 The observation that the existence of invertible solution of a Schröder equation implies embedding into an iteration semigroup is a folklore. Proposition 4.3. Let ϕ : R → R be a real analytic map without critical points. If ϕ [2] has exactly one fixed point u and 0 < ϕ (u) = 1, then ϕ embeds into a real analytic iteration semigroup. In particular, in that case there exist roots of the operator C ϕ of arbitrary order.
Proof. By Proposition 2.7, if ϕ (u) > 1, then the solution f : R → R of the equation C ϕ (f ) = ϕ (u)f is a diffeomorphism. We get the iteration semigroup: Assume that 0 < ϕ (u) < 1. By Theorem 2.6, there is a solution f : R → R of the equation C ϕ (f ) = ϕ (u)f without critical points, f (u) = 0. Since ϕ (u) < 1 then for any x ∈ R the value (ϕ (u)) t · f (x) belongs to the image of f . So we can define the iteration semigroup be the formula (4).
We have completed the proof of Theorem C. Unfortunately, in the case of self maps ϕ with fixed points we cannot characterize when ϕ embeds into a real analytic iteration semigroup. Nevertheless, we suspect that the following conjecture is true.

Conjecture.
A real analytic map ϕ : R → R embeds into a real analytic iteration semigroup if and only if it has no critical point, has at most one fixed point u and in that case 0 < ϕ (u) = 1.
In [8,Section 2] examples of polynomials ϕ without embedding into an iteration semigroup are mentioned. By [34,Th. 15.13], the function ϕ(x) = exp(x)−1 without critical points which has exactly one fixed point u = 0 with ϕ (u) = 1 cannot be embedded into an iteration semigroup. Let us observe that if ϕ embeds into an iteration semigroup Φ then ϕ commutes with every map g t := Φ(t, ·), i.e., g t • ϕ = ϕ • g t . Then the necessary condition for existence of Φ is the existence of many functions commuting with ϕ. This is Vol. 81 (2015) Abel's Functional Equation on Spaces 479 the method used by Baker to show that such Φ does not exist for a wide class of ϕ defined locally around a fixed point or for some meromorphic ϕ; see [34,Th. 10.11] and the papers [2][3][4][5][6][7]44], where a description of analytic functions defined locally around the fixed point and commuting with fixed ϕ is given as well as some criteria of existence of entire functions commuting with some ϕ are given. Especially interesting are papers [5,6] where the case of ϕ with a fixed point u, ϕ (u) = 1, is considered -i.e., the case we cannot decide here. In particular, from [5,Th. 2] it follows that if a real analytic function ϕ : R → R extends to a function meromorphic on C or to an entire function with a fixed point u, ϕ (u) = 1, then the function ϕ cannot be embedded into an iteration semigroup. In [6] more examples related to embeddability into an iteration semigroup are given but mostly they are not real analytic on the whole line.
The case of functions ϕ with two fixed points is considered in [31], again it implies that many functions ϕ cannot be embed into an iteration semigroup. In order to solve the problem of characterization of ϕ which embeds into an iteration semigroup we need to describe global real analytic functions which commute with ϕ. We believe that the question of characterizing ϕ with real analytic iteration roots is even more difficult although also in that case iteration roots of ϕ commute with ϕ so the same necessary condition works.