Operator Theory: Advances and Applications Vol. 144 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences P. Lancaster (Calgary) Tel Aviv University L. E. Lerer (Haifa) Ramat Aviv, Israel B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) Editorial Board: L. Rodman (Williamsburg) J. Arazy (Haifa) J. Rovnyak (Charlottesville) A. Atzmon (Tel Aviv) D. E. Sarason (Berkeley) J. A. Ball (Blacksburg) I. M. Spitkovsky (Williamsburg) A. Ben-Artzi (Tel Aviv) S. Treil (Providence) H. Bercovici (Bloomington) H. Upmeier (Marburg) A. Bottcher (Chemnitz) S. M. Verduyn Lunel (Leiden) K. Clancey (Athens, USA) D. Voiculescu (Berkeley) L. A. Coburn (Buffalo) H. Widom (Santa Cruz) K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville) R. G. Douglas (College Station) D. Yafaev (Rennes) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) Honorary and Advisory B. Gramsch (Mainz) Editorial Board: G. Heinig (Chemnitz) C. Foias (Bloomington) J. A. Helton (La Jolla) P. R. Halmos (Santa Clara) M. A. Kaashoek (Amsterdam) T. Kailath (Stanford) H. G. Kaper (Argonne) P. D. Lax (New York) S. T. Kuroda (Tokyo) M. S. Livsic (Beer Sheva) One-dimensional Functional Equations G. Belitskii V.Tkachenko Springer Basel AG Authors: Genrich Belitskii and Vadim Tkachenko Department of Mathematics Ben Gurion University of the Negev P.O. Box 653 Beer Sheva 84105 Israel 2000 Mathematics Subject Classification 37-xx, 39-xx, 58-xx A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-0348-9431-9 ISBN 978-3-0348-8079-4 (eBook) DOI 10.1007/978-3-0348-8079-4 This work is subject to copyright. AII rights are reserved, whether the whole Of par! of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permis sion of the copyright owner must be obtained. © 2003 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 2003 Softcover reprint ofthe hardcover Ist edition 2003 Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN 978-3-0348-9431-9 Contents Preface ix 1 Implicit Functions 1 1.1 Formal solvability. 2 1.2 Theorem on local solvability. 3 1.3 Transformations of equations 4 1.4 Global solvability ..... 6 1.5 Comments and references . . 7 2 Classification of One-dimensional Mappings 9 2.1 Wandering and non-wandering subsets ...... . .. . . 12 2.2 Mappings with wandering compact sets ... . ..... . 14 2.2.1 Strictly monotonic mappings without fixed points 14 2.2.2 The Abel and cohomological equations . ..... . 16 2.2.3 Smooth and analytic solutions of a cohomological equation 20 2.3 Local structure of mappings at an isolated fixed point 22 2.3.1 Formal classification . 26 2.3.2 Smooth classification. . . . . . . . . 29 2.3.3 Analytic classification . . . . . . . . 39 2.4 Diffeomorphisms with isolated fixed points . 46 2.4.1 Topological classification . . . . . . . 46 2.4.2 Smooth classification of diffeomorphisms with a unique fixed point. . . . . . . . . . . . . . . . . . . . . . . . . .4 7 . . 2.4.3 Smooth classification of diffeomorphisms with several hyperbolic fixed points . . . . . . . . . . . . 51 2.4.4 Another approach to smooth classification . . . . . . . 55 2.5 One-dimensional flows and vector fields ..... .... ... 58 2.5.1 Classification of vector fields in a neighborhood of a singular point. . . . . . . . . . . . . . . . . . . . . . . . . 61. 2.5.2 Flows on the real line with hyperbolic fixed points . . . .. 64 Contents VI 2.6 Embedding problem and iterative roots . .... 68 2.6.1 Mappings without non-wandering points. 68 2.6.2 CO-embedding .. ............ . 68 2.6.3 Diffeomorphisms with a unique fixed point 69 2.6.4 Diffeomorphisms with several fixed points 72 2.7 Comments and references 74 3 Generalized Abel Equation 77 3.1 Local solvability . ...... . ..... ..... ........ 78 3.1.1 Local solvability in a neighborhood of a non-fixed point 78 3.1.2 Proof of Theorem 3.1 for analytic functions 81 3.1.3 Local solvability at an isolated fixed point . . . . . . . . 90 3.1.4 More on analytic solutions . . . . . . . . . . . . . . . 1. 06. 3.2 Global solutions of equations with not more than one fixed point 110 3.2.1 Equations with fixed-point free mappings F . ... . 110 3.2.2 The case of a single fixed point . . . . . . . . . . . . . 113 3.3 Gluing method for linear equations with several fixed points. 120 3.3.1 Cohomological equation . . . . . . . . . 123 3.3.2 Equations with hyperbolic fixed points. 125 3.4 Comments and references . . . . . . . . . . . . 131 4 Equations with Several Transformations of Argument 133 4.1 Local solvability ... 133 4.2 Extension of solutions . . ..... . .. ... . 136 4.2.1 Absorbers ................ . 136 4.2.2 Extension of solutions from an absorber 137 4.2.3 Extension from intersection of absorbers. Decomposition method. . . . . . 139 4.3 Examples ................ . 142 4.4 Difference equations in Carleman classes 147 4.4.1 Decomposition in classes C(mn) 148 4.4.2 Equations with constant coefficients 150 4.4.3 Equations with non-constant coefficients 152 4.5 Comments and references 154 5 Linear Equations 157 5.1 Generalized linear Abel equation . . . . . . . . . . . . . . . 162 5.1.1 Equations on the real line with a unique fixed point 164 5.1.2 Cohomological equation . . . . . . . . . . . . . . . . 167 5.1.3 Spectrum of a weighted shift operator . . . . . . . . 169 5.1.4 Normal solvability of equations with hyperbolic fixed points 172 5.1.5 Equations with periodic points . . . . . . . . . . . . . 17. 7 . . Contents Vll 5.2 Localization of obstacles to solvability ...... . 181 5.3 Equations with constant coefficients . . ..... . 185 5.4 Equation with affine transformations of argument . 190 5.5 Comments and references ............. . 199 Bibliography 201 Index 207 Preface This monograph is devoted to the study of functional equations g(x, <p(x), <p(Fl(X)), ... , <p(Fn(x))) = 0, x E M, (0.1) where M is either the real line lR or the unit circle lI', <p is an unknown function, F1, ... , Fn are given mappings of the manifold M into itself and 9 : M x lRn+ 1 ---> lR is a given mapping. We assume that the mappings f, F1, ... , Fn are of a class Ck, k E NU{ 00, ~}. According to the standard definition CO(M) is the class of continuous functions on M; Ck(M) with 0 < k < 00 consists of k-times continuously differentiable functions on M; n COO(M) = Ck(M) l<k<oo and, in addition, eN is the class of analytic functions on M. The main question related to equation (0.1) is whether it is solvable in one of the above classes. Among the most known particular cases of (0.1) there are the classical equa tion on implicit functions g(x,<p(x)) = 0, x E M, (0.2) which does not contain the transformed argument at all, the Abel equation <p(F(x)) - <p(x) = I, x E M, (0.3) or the more general cohomological equation <p(F(x)) - <p(x) = ')'(x) , x E M, (0.4) and the Schroder equation <p(F(x)) = H(<p(x)), x E M, (0.5) which systematically appears in dynamical problems. x Preface In particular, if M IR and F(x) x + 1, then (0.4) turns out to be a difference equation cp(x + 1) - cp(x) = 'Y(x), x E IR, (0.6) completely investigated long ago. It is well known that for an arbitrary function 'Y there exists a solution cp of (0.6) belonging to the same class Ck as T The general solvability problem for equation (0.1) may be divided in two parts: first, to find conditions which guarantee the existence of its local solution in a neighborhood of some point Xo E M and, second, granted such local solvability at every point x E M, to find additional conditions for the global solvability on M. Let us illustrate this approach on the equation cp(x) = f(x, cp(F(x)), x E IR, (0.7) in classes Ck, 0 ::::: k ::::: 00. If a point Xo E IR is such that F(xo) -I=- xo, then it is easy to construct a local Ck-solution in a neighborhood V of Xo. Namely, let F(V) n V = 0. Then we can choose an arbitrary Ck-function CPo(x),.r, E F(V), define CPl(X) f ( x, CPo ( F ( x ) ) ), x E V, and set cp(x) = {cpo(x), x E F(V), CPl(X), x E V. The function cp may be extended to a Ck-function on a connected neighborhood of the set V U F(V). This yields a local solution since (0.7) is fulfilled for x E V. If Xo is a fixed point of F and Yo = f (xo, Yo), then various fixed point theorems in functional spaces may be applied to construct a local solution. If the mapping F(x) has no more than one fixed point and the Ck-mapping G(x, y) = (F(x), f(x, y)) is Ck-invertible, the above local solutions can be ex tended step-by-step via the equation itself as Ck-solutions on the whole axis R This extension process completely solves the problem of the global solvability modulo local solvability. The approach described above was applied by many authors to the study of rather general functional equations. Numerous results on solvability of such equations in integrable, smooth, analytic, monotonic, convex functions and on the properties of solutions are represented in monographs [53, 54]. Both monographs contain very detailed lists of references. The current state of affairs is described in the survey [12]. Some simple examples show that a local solvability at every point may not imply global solvability. In particular, this is the case when F in (0.7) has more than one fixed point. A similar effect in the theory of differential equations is known as Stocks phenomenon. This is a situation where an equation has local solutions on overlapping domains covering a neighborhood of a point xo, but there are obstacles to gluing them to a solution in a whole neighborhood. Preface Xl Similar obstacles arise for functional equations solvable in a neighborhood of every point x E M and we describe conditions which permit us to glue such local solutions in a global solution on M. The material of the book is organized as follows. Chapter 1 is dedicated to standard questions related to the problem of im plicit functions. In addition to the classical theorem on implicit functions we con sider relations between formal and local solvability and between local and global solvability. In Chapter 2 we investigate equations (0.3)-(0.5) closely related to the prob lem of conjugacy and semi-conjugacy of one-dimensional dynamical systems. Properties of a mapping which are invariant with respect to transformations of variables determine its dynamical behavior. In turn this behavior determines properties of the related functional equations. We study invariants of mappings and find conditions for two given mappings to be conjugate in the corresponding class of smoothness. In Section 1, Chapter 2 we consider the fixed-point free diffeomorphisms of + the real line and prove that they are conjugate with the shift x ---* x 1 and that the conjugating transformation may be chosen from the same smoothness class as the diffeomorphisms themselves. While not very complicated for smooth diffeomorphisms, the proof becomes more technically involved for real-analytic diffeomorphisms. For example, one of these proofs appeals to the uniqueness of the smooth structure on the real line, while another is based on the KAM method. In Section 2, Chapter 2 we investigate equations (0.3) and (0.4) on the real line with arbitrary mappings F, maybe non-invertible. The absence of non wandering points of F is a necessary condition for the solvability of (0.3) in continuous functions. If F is injective, then this condition is sufficient for the solvability not only of (0.3) but also of (0.4) with arbitrary function ')'(x). How ever, if F is not injective, then additional obstacles to the solvability of the Abel equation arise. We introduce a notion of wandering sets for F, and prove that equation (0.4) is solvable in continuous functions for every continuous')' if and only if every compact set is wandering for F. Section 3, Chapter 2 is devoted to a local classification of mappings in a neighborhood of a fixed point. These investigations were initiated by Poincare for analytic functions and by Sternberg for smooth functions. It is well known that if a Ok-mapping F, k 2: 2, has a hyperbolic fixed point xu, then it is reducible to a linear form using a local transformation of the same smoothness, including analytic functions. It is known that this statement is not true for class 01 . The results related to the non-hyperbolic case are less known. The first problem arising here is the problem of a formal classification, i.e., the conjugacy in formal power series. We give a complete list of formal normal forms. Further, we describe a new approach to constructing local invariants in a neighborhood of a fixed point of arbitrary type. This approach is based on the conjugacy of F with a standard shift in the left and right semi-neighborhoods of the fixed point. Using a pair of such straightening