Table Of ContentLecture Notes in Mathematics 1834
Editors:
J.–M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Yosef Yomdin
Georges Comte
Tame Geometry with
Application in Smooth Analysis
1 3
Authors
YosefYomdin
DepartmentofMathematics
WeizmannInstituteofScience
Rehovot76100,Israel
e-mail:[email protected]
GeorgesComte
LaboratoireJ.A.Dieudonne´
UMRCNRS6621
Universite´deNiceSophia-Antipolis
28,avenuedeValrose
06108NiceCedex2,France
e-mail:[email protected]
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MathematicsSubjectClassification(2000):28A75,14Q20,14P10,26B5,26B15,32S15
ISSN0075-8434
ISBN3-540-20612-4Springer-VerlagBerlinHeidelbergNewYork
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Preface
Thisbookpresentsresultsandmethodsdevelopedduringquitealongperiod
of time and many people helped in this work. We would like to thank Y.
Kannai for pointing out, in the very beginning of the work on quantitative
transversality,therelevanceofmetricentropy.Since1983M.Gromovencou-
raged this research and helped us in many fruitful discussions of qualitative
transversality and Semialgebraic Geometry in Dynamics and Analysis. We
would like to thank him especially for suggesting a problem of quantitative
Kupka-Smale, for his contribution to Ck-reparametrization of semialgebraic
setsandapplicationstodynamics,forprovidingacentral(forthisbook)refe-
rence to Multidimensional Variations and to books of Vitushkin and Ivanov
and for encouraging writing preliminary texts, which were used in this book.
This book would not have be written without the help and encouragement
of J. -J. Risler and B. Teissier and numerous fruitful discussions with them
duringallthelongperiodofthebook’spreparation.Itisapleasuretothank
M. Giusti, J. -P. Henry and M. Merle for their invitation to give a course
on Metric Semialgebraic Geometry at E´cole Polytechnique in 1985-86, and
againM.MerleforhisremarksduringlecturesgivenattheUniversityofNice
- Sophia Antipolis in 1999-2000, and P. Milman for fruitful discussions and
foraninvitationtotheUniversityofToronto,wherethepreliminarytexthas
been written and typed. We would like to thank D. Trotman, who read and
corrected the text, and also indicated precious references. We would like to
thank M. Briskin, Y. Elichai, J.-P. Franc¸oise, G. Loeper and N. Roytvarf for
their help and contribution. Many of the results and methods presented in
this book have been obtained in a long collaboration with them.
Lastnotleast,manythanksbelongtotheUniversityofBeer-Sheva,tothe
Max Planck Institut fu¨r Mathematik, to IHES, to the University of Toronto
and to our home Universities - the University of Nice-Sophia Antipolis and
the Weizmann Institute of Science, where many of the results and methods
in this book have been developed, and to the Israel Science Foundation and
the Minerva Foundation, which have supported the research of one of the
authors for many years.
Table of Contents
Preface ....................................................... V
Table of Contents............................................. VII
1 Introduction and Content................................. 1
1.1 Motivations............................................ 1
1.2 Content and Organization of the Book .................... 4
1.3 The Motion Planing Problem in Robotics as an Example .... 7
1.4 A Proof of the Morse-Sard Theorem in the Simplest Case.... 18
2 Entropy .................................................. 23
3 Multidimensional Variations .............................. 33
4 Semialgebraic and Tame Sets ............................. 47
5 Variations of Semialgebraic and Tame Sets ............... 59
6 Some Exterior Algebra ................................... 75
7 Behaviour of Variations under Polynomial Mappings ..... 83
8 Quantitative Transversality and Cuspidal Values.......... 99
9 Mappings of Finite Smoothness........................... 109
10 Some Applications and Related Topics ................... 131
10.1 Applications of Quantitative Sard
and Transversality Theorems............................. 132
10.1.1 Maxima of smooth families ........................ 132
10.1.2 Average topological complexity of fibers............. 133
10.1.3 Quantitative Kupka-Smale Theorem ................ 134
10.1.4 Possible Applications in Numerical Analysis ......... 136
10.2 Semialgebraic Complexity of Functions.................... 148
10.2.1 Semialgebraic Complexity ......................... 149
10.2.2 Semialgebraic Complexity and Sard Theorem ........ 152
10.2.3 Complexity of Functions
on Infinite-Dimensional Spaces ..................... 153
VIII Table of Contents
10.3 Additional Directions ................................... 155
10.3.1 Asymptotic Critical Values of Semialgebraic
and Tame Mappings .............................. 155
10.3.2 Morse-Sard Theorem in Sobolev Spaces ............. 156
10.3.3 From Global to Local: Real Equisingularity.......... 157
10.3.4 Ck Reparametrization of Semialgebraic Sets.......... 158
10.3.5 Bernstein-Type Inequalities for Algebraic Functions... 159
10.3.6 Polynomial Control Problems ...................... 161
10.3.7 Quantitative Singularity Theory.................... 165
Glossary...................................................... 171
References.................................................... 173
1 Introduction and Content
1.1 Motivations
This book deals with several related topics:
– Geometryofrealsemialgebraicandtamesets(i.e.setsdefinableinsomeo-
minimalstructureon(R,+,.)),withthestresson“metric”characteristics:
lengths, volumes in different dimensions, curvatures etc...
– Behaviour of these characteristics under polynomial mappings.
– Integral geometry, especially the so-called Vitushkin variations, with the
stress on applications to semialgebraic and tame sets.
– Geometry of critical and near critical values of differentiable mappings.
Some fractal geometry naturally arising in this context.
Below we give a short description of each of these topics, their mutual
dependance and logical order. Motivation for the type of question asked in
this book, comes from several different sources: the main ones are Differen-
tial Topology, Singularity Theory, Smooth Dynamics, Control, Robotics and
Numerical Analysis.
One of the main analytic results, underlying most basic constructions of
Differential Analysis, Differential Topology, Differential Geometry, Differen-
tial Dynamics, Singularity Theory, as well as nonlinear Numerical Analysis,
is the so-called Sard (or Morse-Sard, or Morse-Morse-Sard, see [Morse 1,2],
[Mors], [Sar 1-3]) theorem. It asserts that the set of critical values of a suffi-
cientlysmoothmappinghasmeasurezero.Mostlythistheoremappearsasan
assumption,thatthesetY(c)ofsolutionsofanequationf(x)=cisasmooth
submanifoldofthedomainofx,foralmostanyvaluecoftherighthandside
(in the semialgebraic or tame case, an asymptotic version of the Morse-Sard
theorem(see[Rab],[Kur-Orr-Sim])showsthatf inducesafibrationoverthe
connected components of the “good values” c).
Another typical appearence of the Morse-Sard theorem is in the form of
varioustransversalitystatements:byasmallperturbationofthedatawecan
always achieve a situation where all the submanifolds of interest intersect
one another in a transversal way. Fig.1.1 shows a non-transversal (a) and
transversal (b, c) intersections of two plane curves.
Y.YomdinandG.Comte:LNM1834,pp.1–22,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Introduction and Content
Fig. 1.1.
Technically, the Morse-Sard theorem is a rather subtle fact. It is well
known since the classical examples of Whitney [Whi 1], that the geometry of
the critical points and of the critical values of a differentiable function can
be very complicated. In particular, the requirement that the function must
haveatleastthesamenumberofderivativesasthedimensionofthedomain,
cannotberelaxed.Andusuallyanalyticfactsthatincorporateinanessential
way the existence and the properties of the high order derivatives, are deep,
both conceptually and technically.
However, in classical applications the Morse-Sard theorem appears as a
background fact, as a default assumption, that all the transversality and
regularity properties required can be achieved by small perturbations of the
data.Theinterplaybetweenthehighorderanalyticstructureofthemappings
involved and their geometry rarely becomes apparent. The main reason is
that the classical Morse-Sard theorem is basically qualitative. Its conclusion
appears as an “existential” fact and it provides no quantitative information
on the solutions, submanifolds etc... in question.
Averynaturalquantitativesettingoftheproblems,coveredbytheMorse-
Sard theorem, is possible. In the case of transversality, the question is: given
a maximal size of perturbations allowed, how strong a transversality of the
perturbed submanifolds can be achieved? (For plane curves the transversality
can be measured just by the angle between the curves at their intersection
points.Fig.1.1cpresentsastrongtransversality,incontrasttoaweakonein
Fig.1.1b).
Concerning the set Y(c) of solutions of the equation f(x) = c, a natural
quantitative question is: How much of the complexity of Y(c) (geometric or
topological)canbeeliminatedbyaperturbationofcwithintheallowedrange?
What is the average complexity (with respect to c) of Y(c)?
In nonlinear Numerical Analysis, a typical conclusion, provided by the
classical Morse-Sard theorem, is that with probability one certain determi-
nants do not vanish. However, to organize computations in a stable way it is
necessary to get a quantitative information: how big are these determinants
usually ?
The importance of this sort of quantitative information has been realized
during the last decades in many fields. In the study of the complexity of
