Introduction to the qualitative theory of dynamical systems on surfaces:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
Providence, RI
American Math. Soc.
1996
|
| Series: | Translations of mathematical monographs
153 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007404808&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | PST: Vvedenie v kačestvennuju teoriju dinamičeskich sistem na poverchnostjach. - Aus dem russ. Ms. übers. |
| Physical Description: | XIII, 325 S. graph. Darst. |
| ISBN: | 0821803697 |
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| 245 | 1 | 0 | |a Introduction to the qualitative theory of dynamical systems on surfaces |c S. Kh. Aranson ; G. R. Belitsky ; E. V. Zhuzhoma |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1996 | |
| 300 | |a XIII, 325 S. |b graph. Darst. | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 153 | |
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Record in the Search Index
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| adam_text | Contents
Foreword xiii
Chapter 1. Dynamical Systems on Surfaces 1
§1. Flows and vector fields 1
1.1. Definitions and examples 1
1.2. Connection between flows and vector fields 2
1.3. Vector fields and systems of differential equations 3
1.4. Diffeomorphisms of vector fields 3
§2. Main ways of specifying flows on surfaces 3
2.1. The projection method 4
2.2. Systems of differential equations in local charts 6
2.3. Specification of a flow with the help of a universal covering 7
2.3.1. Transformation groups 8
2.3.2. Flows on the torus 8
2.3.3. Flows on closed orientable surfaces of genus 1 10
2.4. Specification of a flow with the help of a branched covering 12
2.4.1. Definition of a branched covering 12
2.4.2. Covering flows 12
2.4.3. Construction of transitive flows 13
2.5. The pasting method 15
2.6. Suspensions 16
2.6.1. The suspension over a homeomorphism of the circle 16
2.6.2. The suspension over an exchange of open intervals 17
2.7. Whitney s theorem 21
2.7.1. The theorem on continuous dependence on the initial
conditions 21
2.7.2. The rectification theorem 22
2.7.3. Orientability 22
§3. Examples of flows with limit set of Cantor type 23
3.1. The example of Denjoy 24
3.2. Cherry flows 27
3.3. An example of a flow on the sphere 31
§4. The Poincare index theory 34
4.1. Contact free segments and cycles 34
4.2. The index of a nondegenerate cycle in a simply connected do¬
main 35
4.3. The index of an isolated equilibrium state 36
4.4. The Euler characteristic and the Poincare index 38
vii
viii CONTENTS
4.5. Connection between the index and the orientability of foliations 38
4.6. An example of a foliation that is locally but not globally ori¬
ent able 40
Remark. About a result of El sgol ts 43
Chapter 2. Structure of Limit Sets 45
§1. Initial concepts and results 45
1.1. The long flow tube theorem, and construction of a contact free
cycle 45
1.2. The Poincare mapping 47
1.3. The limit sets 48
1.4. Minimal sets 51
1.5. Nonwandering points 52
§2. The theorems of Maier and Cherry 53
2.1. Definitions of recurrence 53
2.2. The absence of nontrivial recurrent semitrajectories on certain
surfaces 55
2.3. The Cherry theorem on the closure of a recurrent semitrajectory 57
2.4. The Maier criterion for recurrence 61
2.5. The Maier estimate for the number of independent nontrivial
recurrent semitrajectories 65
§3. The Poincare Bendixson theory 66
3.1. The Poincare Bendixson theorem 67
3.2. Bendixson extensions 67
3.3. Separatrices of an equilibrium state 70
3.4. The Bendixson theorem on equilibrium states 74
3.5. One sided contours 77
3.6. Lemmas on the Poincare mapping 78
3.7. Description of quasiminimal sets 83
3.8. Catalogue of limit sets 85
3.9. Catalogue of minimal sets 87
§4. Quasiminimal sets 87
4.1. An estimate of the number of quasiminimal sets 88
Remarks. The estimates of Aranson, Markley, and Levitt 89
4.2. A family of special contact free cycles 89
4.3. Partition of a contact free cycle 90
4.4. The Gardiner types of partition elements 91
4.5. The structure theorem 96
Chapter 3. Topological Structure of a Flow 101
§1. Basic concepts of the qualitative theory 101
1.1. Topological and smooth equivalence 101
1.2. Invariants 103
1.3. Classification 104
§2. Decomposition of a flow 104
2.1. Characteristic curves of a quasiminimal set 104
2.2. Periodic elements of a partition 106
2.3. Criterion for a flow to be irreducible 109
CONTENTS ix
2.4. Decomposition of a flow into irreducible flows and flows without
nontrivial recurrent semitrajectories 110
2.5. The Levitt decomposition 112
Remark 1. The canonical decomposition 116
Remark 2. The center of a flow 117
§3. The structure of an irreducible flow 117
3.1. Blowing down and blowing up operations 117
3.2. Irreducible flows on the torus 122
§4. Flows without nontrivial recurrent trajectories 125
4.1. Singular trajectories 125
4.2. Cells 127
4.3. Topology of cells 127
4.4. Structure of a flow in cells 131
4.5. Smooth models 132
4.6. Morse Smale flows 136
4.7. Cells of Morse Smale flows 139
§5. The space of flows 139
5.1. The metric in the space of flows 139
5.2. The concepts of structural stability and the degree of structural
instability 140
5.3. The space of structurally stable flows 141
5.4. Flows of the first degree of structural instability 143
5.5. On denseness of flows of the first degree of structural instability
in the space of structurally unstable flows 146
Chapter 4. Local Structure of Dynamical Systems 147
§1. Dynamical systems on the line 148
1.1. Linearization of a diffeomorphism 148
1.2. Lemmas on functional equations 149
1.3. Proof of Theorem 1.1 153
1.4. Flows on the line 154
§2. Topological linearization on the plane 156
2.1. Formulation of the theorem 156
2.2. Proof of the theorem 156
§3. Invariant curves of local diffeomorphisms 157
3.1. Invariant curves of a node 158
3.2. Invariant curves of a saddle point 159
§4. C1 linearization on the plane 160
§5. Formal transformations 163
5.1. Formal mappings 163
5.2. Conjugacy of formal mappings 164
5.3. Formal vector fields and flows 166
§6. Smooth normal forms 168
6.1. Normal forms with flat residual 168
6.2. Smooth normal forms of a node 169
6.3. Smooth normalization in a neighborhood of a saddle point 172
6.4. The Sternberg Chern theorem 176
6.5. The smoothness class as an obstacle to smooth normalization 178
§7. Local normal forms of two dimensional flows 179
x CONTENTS
7.1. Topological and (^ linearization 179
7.2. Invariant curves of a flow 180
7.3. Smooth normal forms 181
7.4. The correspondence mapping at a saddle point 183
§8. Normal forms in a neighborhood of an equilibrium state (survey
and comments) 184
Chapter 5. Transformations of the Circle 189
§1. The Poincare rotation number 189
1.1. Definitions and notation 189
Remark 1. The rotation set of a continuous transformation of
degree 1 193
Remark 2. The rotation set of a topological Markov chain 195
Remark 3. The rotation set of a mapping of Lorenz type 195
1.2. Invariance of the rotation number 196
1.3. Continuous dependence of the rotation number on a parameter 197
1.4. The rotation number of a homeomorphism of the circle 198
§2. Transformations with irrational rotation number 199
2.1. Transformations semiconjugate to a rotation 199
2.2. A criterion for being conjugate to a rotation 202
2.3. Limit sets 203
2.4. Classification of transitive homeomorphisms 204
2.5. Classification of Denjoy homeomorphisms 205
2.6. Classification of Cherry transformations 209
§3. Structurally stable diffeomorphisms 213
3.1. The Cr topology 213
3.2. Main definitions 213
3.3. Instability of an irrational rotation number 215
3.4. Openness and denseness of the set of weakly structurally stable
diffeomorphisms 217
3.5. Classification of weakly structurally stable diffeomorphisms 217
Remark. Diffeomorphisms of the first degree of structural insta¬
bility 217
§4. The connection between smoothness properties and topological
properties of transformations of the circle 218
4.1. Continued fractions 218
4.2. The order of the points on the circle 220
4.3. The theorem of Denjoy 221
4.4. The theorem of Yoccoz 223
4.5. Corollary to the theorem of Yoccoz for Cherry transformations 227
4.6. The Herman index of smooth conjugacy to a rotation 227
§5. Smooth classification of structurally stable diffeomorphisms 230
5.1. Pasting cocycles 230
5.2. C° Q conjugacy 232
5.3. Smooth classification 234
5.4. Corollaries 235
5.5. Conjugacy of flows 236
5.6. Inclusion of a diffeomorphism in a flow 238
5.7. Comments 238
CONTENTS xi
Chapter 6. Classification of Flows on Surfaces 239
§1. Topological classification of irreducible flows on the torus 239
1.1. Preliminary facts 239
1.2. Curvilinear rays 244
1.3. Asymptotic directions 245
1.4. The Poincare rotation number 249
1.5. The rotation orbit 251
1.6. Classification of minimal flows 253
1.7. Classification of Denjoy flows 254
Appendix. Polynomial Cherry flows 259
§2. The homotopy rotation class 262
2.1. Lobachevsky geometry and uniformization 262
2.2. The axes of hyperbolic isometries 263
2.3. Asymptotic directions 265
2.4. Arithmetic properties of the homotopy rotation class 268
2.5. The homotopy rotation class of a nontrivial recurrent semitra
jectory 270
2.6. The connection between quasiminimal sets and geodesic lami¬
nations 273
2.7. Accessible points of the absolute 279
2.8. Classification of accessible irrational points 280
2.9. The orbit of a homotopy rotation class 283
§3. Topological equivalence of transitive flows 285
3.1. Homotopic contact free cycles 285
3.2. Auxiliary results 286
3.3. Construction of a fundamental domain 288
3.4. Necessary and sufficient conditions for topological equivalence
of transitive flows 290
Remark. Levitt s counterexample to a conjecture of Katok 293
§4. Classification of nontrivial minimal sets 294
4.1. Special and basic trajectories 294
4.2. The canonical set 295
4.3. Topological equivalence of minimal sets 297
4.4. Realization of nontrivial minimal sets by geodesic curves 299
§5. Topological equivalence of flows without nontrivial recurrent tra¬
jectories 300
5.1. Schemes of semicells 300
5.2. Schemes of spiral cells 302
5.3. The orbit complex 303
5.4. Neighborhoods of limit singular trajectories 305
5.5. Main theorems 307
Chapter 7. Relation Between Smoothness Properties and Topological
Properties of Flows 309
§1. Connection between smoothness of a flow and the existence of a
nontrivial minimal set 309
1.1. The theorems of Denjoy and Schwartz 309
1.2. The theorem of Neumann 310
1.3. The theorem of Gutierrez 315
xii CONTENTS
§2. The problem of Cherry 315
2.1. Gray and black cells 315
2.2. The Poincare mapping in a neighborhood of a structurally stable
saddle 316
2.3. Sufficient conditions for the absence of gray cells 318
2.4. Cherry flows with gray cells 318
Bibliography 321
|
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| id | DE-604.BV011057066 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:05:32Z |
| institution | BVB |
| isbn | 0821803697 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007404808 |
| oclc_num | 34669346 |
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| owner_facet | DE-12 |
| physical | XIII, 325 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spellingShingle | Aranson, S. Ch Belickij, Genrich R. 1941- Žužoma, Evgenij V. 1951- Introduction to the qualitative theory of dynamical systems on surfaces Translations of mathematical monographs Dynamique différentiable ram Flows (Differentiable dynamical systems) Stochastische Strömung (DE-588)4195753-2 gnd Niederdimensionales System (DE-588)4202325-7 gnd |
| subject_GND | (DE-588)4195753-2 (DE-588)4202325-7 |
| title | Introduction to the qualitative theory of dynamical systems on surfaces |
| title_auth | Introduction to the qualitative theory of dynamical systems on surfaces |
| title_exact_search | Introduction to the qualitative theory of dynamical systems on surfaces |
| title_full | Introduction to the qualitative theory of dynamical systems on surfaces S. Kh. Aranson ; G. R. Belitsky ; E. V. Zhuzhoma |
| title_fullStr | Introduction to the qualitative theory of dynamical systems on surfaces S. Kh. Aranson ; G. R. Belitsky ; E. V. Zhuzhoma |
| title_full_unstemmed | Introduction to the qualitative theory of dynamical systems on surfaces S. Kh. Aranson ; G. R. Belitsky ; E. V. Zhuzhoma |
| title_short | Introduction to the qualitative theory of dynamical systems on surfaces |
| title_sort | introduction to the qualitative theory of dynamical systems on surfaces |
| topic | Dynamique différentiable ram Flows (Differentiable dynamical systems) Stochastische Strömung (DE-588)4195753-2 gnd Niederdimensionales System (DE-588)4202325-7 gnd |
| topic_facet | Dynamique différentiable Flows (Differentiable dynamical systems) Stochastische Strömung Niederdimensionales System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007404808&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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