Table Of ContentOperator Theory: Advances and
Applications
Vol. 144
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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
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© 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
