Reversibility in Dynamics and Group Theory

296 Pages·2015·1.303 MB·English

by  Anthony G. O'Farrell, Ian Short

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:
ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UnitedKingdom
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LondonMathematicalSocietyLectureNoteSeries:416
Reversibility in Dynamics and
Group Theory
ANTHONY G. O’FARRELL
NationalUniversityofIreland,Maynooth
IAN SHORT
TheOpenUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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©A.G.O’FarrellandI.Short2014
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Firstpublished2014
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Contents
Preface pageix
1 Origins 1
1.1 Originsindynamicalsystems 1
1.2 Originsinfinitegrouptheory 5
1.3 Originsinhiddendynamics 9
1.4 Thereversibilityproblem 15
2 Basicideas 22
2.1 Reversibility 22
2.2 Reformulations 32
2.3 Signedgroups 34
2.4 Whithernext? 34
3 Finitegroups 39
3.1 Reversersoffiniteorder 39
3.2 Dihedralgroups 40
3.3 Symmetricgroups 40
3.4 Alternatinggroups 42
3.5 Groupcharacters 46
3.6 Charactersandreversibleelements 49
3.7 Charactersandstrongly-reversibleelements 50
3.8 Examples 52
3.9 Freegroups 54
4 Theclassicalgroups 59
4.1 Theclassicalgroups 59
4.2 Thegenerallineargroup 60
4.3 Theorthogonalgroup 68
4.4 Theunitarygroup 71
4.5 Summary 75
v
vi Contents
5 Compactgroups 81
5.1 Reversibilityincompactgroups 81
5.2 CompactLiegroups 82
5.3 Thespecialunitarygroup 83
5.4 Compactsymplecticgroups 83
5.5 Thespinorgroups 87
6 Isometrygroups 92
6.1 Isometriesofspherical,Euclidean,andhyperbolicspace 92
6.2 Hyperbolicgeometryintwoandthreedimensions 92
6.3 Euclideanisometries 95
6.4 Hyperbolicisometries 98
7 Groupsofintegermatrices 105
7.1 Conjugacytorationalcanonicalform 106
7.2 Integralquadraticforms 108
7.3 Conway’stopograph 112
7.4 Gauss’methodfordefiniteforms 117
7.5 EllipticelementsofGL(2,Z) 118
7.6 Centralisers 122
7.7 Reversibleelements 123
7.8 Secondassociatedform 128
7.9 Gauss’methodforindefiniteforms 131
7.10 Anotherway 135
7.11 Cyclically-reducedwords 137
8 Realhomeomorphisms 145
8.1 Involutions 146
8.2 Conjugacy 147
8.3 Reflectionalandrotationalsymmetries 154
8.4 Reversibleelements 158
8.5 Productsofinvolutionsandreversibleelements 162
9 Circlehomeomorphisms 166
9.1 Involutions 167
9.2 Conjugacy 168
9.3 Reversibleelements 172
9.4 Strongly-reversibleelements 173
9.5 Productsofinvolutions 183
10 Formalpowerseries 188
10.1 Powerseriesstructures 188
10.2 Elementsoffiniteorder 192
Contents vii
10.3 Conjugacy 193
10.4 Centralisers 196
10.5 ProductsofinvolutionsmoduloX4 203
10.6 Strongly-reversibleelements 204
10.7 Productsofinvolutions 206
10.8 Reversibleseries 208
10.9 Reversers 210
10.10 Productsofreversibleseries 211
11 Realdiffeomorphisms 216
11.1 Involutions 217
11.2 Fixed-point-freemaps 218
11.3 Centralisers 220
11.4 Reversibilityoforder-preservingdiffeomorphisms 223
11.5 Reversibilityinthefulldiffeomorphismgroup 226
11.6 Productsofinvolutionsandreversibleelements 230
12 Biholomorphicgerms 234
12.1 Elementsoffiniteorder 235
12.2 Conjugacy 236
12.3 E´calle–Voronintheory 237
12.4 Roots 242
12.5 Centralisersandflows 243
12.6 Reversibleandstrongly-reversibleelements 245
12.7 Theorderofareverser 247
12.8 Examples 248
References 261
Listoffrequentlyusedsymbols 275
Indexofnames 277
Subjectindex 279