Germs of Diffeomorphisms in the Plane
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902 Freddy Dumortier Paulo R. Rodrigues Robert Roussarie
Germs of Diffeomorphisms in the Plane
Springer-Verlag Berlin Heidelberg New York 1981
Authors
Freddy Oumortier Limburgs Universitair Centrum, Universitaire Campus B-361O Oiepenbeek, Belgium Paulo R. Rodrigues Oepartemento de Geometria, Instituto de Matematica Universidade Federal Fluminense 24000 Niteroi, Brazil Robert Roussarie Departement de Mathematique, Universite de Oijon - UER MIPC Laboratoire de Topologie ERA No.945 du CNRS, 21000 Oijon, France
AMS Subject Classifications (1980): 34C25, 34010, 34030, 58F10, 58F14, 58F22, 58F30
ISBN 3-540-11177-8 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11177-8 Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
TABLE OF CONTENTS - Summary, some motivation and acknowledgments • • • • • • . . • • - Chapter I
Introduction, definitions, formal study and statement of the results
8
Introduction • • • • • • • 2
The blowing-up method
14
3
Statement of the fundamental theorem
20
4
Decomposition in sectors for singularities of vector fields in IR
§
7
5
2
and characteristic lines
22
Statement of the results concerning characteristic lines and decomposition in sectors for certain germs of planar diffeomorphisms; reduction to the fundamental theorem • • • • • • • 00
26
6
Statement of the principal C results
34
7
Statement of the topological results
39
8
Some applications and examples • • •
43
- Chapter II : Stability of type 1- and type 11- singularities.
48
Singularities of the "hyperbolic contraction"-type • • ••
48
2
Singularities of the "quasi-hyperbolic contraction"-type.
51
3
Singularities which are quasi-hyperbolic contractions with respect to a degenerate Finsler-metric
64
4
The "attracting corner"-singularities ••
74
5
Attracting arcs. • • • •
79
6
"Saddle-type"-corners.
80
IV
- Chapter III : Stability of type III-singularities •• Simplified form of the "type III-singularities" • • • • • 2 -Existence of a COO center manifold • • • 00
3
Reduction of the C problem to a formal problem
4
Reduction of the formal problem (30) to a difference equation • •
§
5
86 105
Resolution of the difference equations (39)
113
- Chapter IV : Proof of the COO results • • • • • • • • • • 1
84
108
and (40) • • •
§
83
126
00
On the unicity of germs of flat C diffeomorphisms commuting with attracting diffeomorphisms of type
127
I and II §
2
00
Characterization of germs of flat C diffeomorphisms commut
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