Finiteness Theorems for Limit Cycles

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Finiteness Theorems for Limit Cycles

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     ©1991 by the American Mathematical Society.

     ISBN 0-8218-4553-5

     QA37114413 1991 515'.35-dc2O

     LCCN 91-27411

Dedication

Contents

Foreword

Introduction

     §0.1. Formulation of results: finiteness theorems and the identity theorem

          A. Main theorems

          B. Reductions: A geometric lemma

          C. The reduction: Resolution of singularities

     §0.2. The theorem and error of Dulac

          A. Semiregular mappings and the theorem of Dulac.

          B. The lemma of Dulac and a counterexample to it

          C. Monodromy transformations with nonzero flat correction

          D. The scheme for proving the identity theorem.

          E. The classification theorem.

          F. The scheme for proving Dulac's theorem, and the correspondence mappings.

          G. The correspondence mapping for a hyperbolic sector of a saddle node ordegenerate saddle

          H. Conclusion of the proof of Dulac's theorem

     §0.3. Finiteness theorems for polycycles with hyperbolic vertices

          A. Almost regular mappings

          B. Going out into the complex plane, and the proof of Theorem IV bis

          C. Hyperbolicity and almost regularity

          D. The second geometric lemm

     §0.4. Correspondence mappings for degenerate elementary singular points. Normalizing cochains

          A. Formulations

          B. Proof of the theorem on the correspondence mapping

          C. Proof of the supplement to the theorem on sectorial normalization

          D. The realness of the derivative g'(0) in the expression for the correspondence mapping of a degenerate elementary singular point (a supplement to Theorem 2 in A).

     §0.5. Superexact asymptotic series

     §0.6. Historical comments

CHAPTER I  Decomposition of a Monodromy Transformationin to Terms with Noncomparable Rates of Decrease

     §1.1. Functional cochains and map-cochains

     §1.2. Transition to the logarithmic chart. Extension of normalizing cochains

     §1.3. The composition characteristic, proper choice of semitransversal, and the first step in decomposition of a monodromy transformatio

     §1.4. Multiplicatively Archimedean classes. Heuristic arguments

          A. Classes of Archimedean equivalence

          B. Proper groups and the Archimedean classes corresponding to them

          C. Archimedean classes corresponding to a proper group

          D. Motivations for the basic definitions

     §1.5. Standard domains and admissible germs of diffeomorphisms

     §1.6. Germs of regular map-cochains, RROK

          A. Regular partitions

          B. Regular cochains.

          C. Special sets of admissible germs

          E. Equivalent and negligible diffeomorphisms

     §1.7. Main definitions: standard domains, superexact asymptotic series, and regular functional cochains of class n

     §1.8. The multiplicative and additive decomposition theorems

     §1.9. Reduction of the finiteness theorem to auxiliary results

     §1.10. Group properties of regular map-cochains

     §1.11. Proofs of the decomposition theorems

          A. Proof of the multiplicative decomposition theorem, MDTn

          B. The proof of Proposition 1

          C. The proof of the additive decomposition theorem ADT

          D. A criterion for being weakly real

          E. Conclusion of the proof of the additive decomposition theorem.

CHAPTER II  Function-Theoretic Properties of Regular Functional Cochains

     §2.1. Differential algebras of cochains

     §2.2. Completeness

     §2.3. Shifts of functional cochains by slow germs

     §2.4. The first shift lemma, regularity: SL i, reg

          A. Formulation of the basic lemma and the auxiliary lemma

          B. Conclusion of the proof of the first shift lemma (regularity).

          C. Proof of Lemma 5.21

     §2.5. Shifts of cochains by cochains

     §2.6. Properties of special admissible germs of class n

     §2.7. The second shift lemma and the conjugation lemma, function-theoretic variant: SL 2n, reg and CLn, reg

          A. Formulations

          B. Induction step: implication A.

          C. Group properties of special admissible germs

          D. Estimate of the germ p.

          E. Implication B.

     §2.8. The third shift lemma, function-theoretic variant, SL 3n, reg

     §2.9. Properties of the group A - n J n -1

     §2.10. The fourth shift lemma, assertion a

     § 2.11. The fourth and fifth shift lemmas, function-theoretic variant

          A. Reduction to the properties of the group Lm

          B. Group properties of the sets Ln and "o

          C. Properties of the sets Ln -1 and yn -1

          D. Group properties of the sets Ln -1 and o1 -1

     §2.12. The Regularity Lemma

CHAPTER III  The Phragmen-Lindel of Theoremfor Regular Functional Cochains

     §3.1. Classical Phragmen-Lindelof theorems and modifications of them

     §3.2. A preliminary estimate and the scheme for proving the Phragmen-Lindelof theorem

          A. Heuristic arguments

          B. A preliminary estimate.

          C. The derivation of the Phragmen-Lindelof theorem from the preliminary estimates

     §3.3. Proper conformal mappings

          A. Definitions.

          B. Examples.

          C. Properties of proper mappin

     § 3.4. Trivialization of a cocycle

          A. Properties of regular partitions

          B. Lemma on trivialization

     §3.5. The maximum principle for functional cochains

     §3.6. A general preliminary estimate

     §3.7. A preliminary estimate of cochains given in a standard domain

     §3.8. The Warschawski formula and corollaries of it

     § 3.9. A preliminary estimate for cochains given in domains corresponding to rapid germs

          A. Construction of the domain

          B. The conformal mapping  , and a proof that the preliminaryestimate is effective in the domai

          C. Effectiveness of the preliminary estimate

          D. The preliminary estimate in the case when the germ anext is rapid.

          E. Proof of the preliminary estimate II in the case when the germ a is rapidand anext is sectorial.

          F. Properness of the mapping

          G. Proof of Proposition 1 in subsection

     §3.10. Preliminary estimates in domains corresponding to sectorial and slow germs

          A. Construction of domains corresponding to germs that are not

          B. The conformal mapping

          C. Effectiveness of the preliminary estimate

          D. Proof that yr is proper.

          E. Investigation of the estimate given by Lemma 4: the function

          F. Investigation of the estimate given by Lemma 4: proof of the preliminary estimate II.

CHAPTER IV  Superexact Asymptotic Series

     §4.1. The induction hypothesis

     §4.2. Multiplication Lemma MLn

     §4.3. First Shift Lemma, SL

          A. The group G

          B. Properness of the group G

          C. Shifts of functional cochains of class k by elements of the group G`nfork n - 2.

          F. The action of the operator A -1 on germs with slowly decreasing correction.

          G. Corollaries

     §5.3. Ordering, holomorphicity, convexity

          A. Exponential extensions of ordered algebras

          B. REMARK

          C. The connection between ordering, inclusion, and convexity.

          D. Small perturbations, definiteness of sign, and convexity

     § 5.4. Standard domains of class n + 1 and admissible germs of diffeomorphisms comparable with linear germs

          .A. Formulations

          B. A criterion for domains to be standard, and the property of being standard for domains of class n + 1 .

          C. Boundedness of the derivatives and Property 1 of standard domains

          D. Compositions of mappings whose corrections have real parts of definite sign

          E. Domains of class n + 1 and ordered differential algebras

          G. The properties of standard domains of class n + 1 : continuation

          H. Germs of class

          I. Admissibility of germs of class

          J. Estimates of the real parts of germs of class

     §5.5. Admissible germs increasing more rapidly than linear germs

          A. Formulations

          B. Germs of class osnlo w1 with generalized exponent 1 .

          C. Boundedness of derivatives

          D. Mappings of standard domains

          E. An estimate of the real parts of germs of class (oSlo 1) -1 in standard domains of class n + 1 .

     § 5.6. Admissible germs of class

     §5.7. Proof of Lemma 5.7n+1

          A. Formulation

          B. Connection between the geometric and analytic ordering of special admissible germs

          C. Half-strips of type W.

          D. Boundary functions of special half-strips

          E. Proof of Proposition 1

          F. Proof of Proposition 2.

     § 5.8. Convexity

          A. Comparison of a harmonic function of constant sign with its derivatives

          B. The convexity lemma for slow germs of c

          C. Proof of Proposition 2.

          D. Arguments of derivatives of germs in the class

          E. The convexity lemma for germs of clas

          F. Estimate of the argument of the derivative of a germ of class

     §5.9. Distortion theorems for special admissible germs

          A. Formulations

          B. Slow germs that are not comparable with linear germs

          C. Almost linear germ

          D. Proof of Lemma 5.8n+1 in the case of sectorial and rapid germ

Bibliography

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