Ergodic theory of fibred systems and metric number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford
Clarendon Press [u.a.]
1995
|
| Series: | Oxford science publications
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 295 S. |
| ISBN: | 0198534884 |
Staff View
MARC
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| 650 | 4 | |a Ergodic theory | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Basic definitions 1
1.1 Fibred systems 1
1.2 Cylinders 3
2 Series expansions 7
2.1 p adic transformation 7
2.2 /3 adic transformation 7
2.3 Balkema Oppenheim series 9
2.4 Alternating series 11
2.5 Infinite products 11
3 Continued fractions 13
3.1 Simple (or regular) continued fractions 13
3.2 Continued fractions to the nearest integer 14
3.3 Odd and even partial quotients 17
3.4 Hirzebruch s continued fractions 19
3.5 Japanese continued fractions. I 19
3.6 Japanese continued fractions. II 23
3.7 A continued fractions 24
4 Algorithms of cotangent type 25
4.1 Lehmer s cotangent algorithm 25
4.2 The algorithm of F. Ryde 26
5 Unimodal and other maps 27
5.1 Unimodal maps 27
5.2 Piecewise property .E maps 28
5.3 S unimodal maps 29
ix
x CONTENTS
6 Multidimensional continued fractions 31
6.1 Brun s algorithm 31
6.2 Selmer s algorithm 33
6.3 Jacobi Perron algorithm 35
6.4 Giiting s algorithm 36
6.5 Skew products 37
6.6 Generalized subtractive algorithm 37
6.7 Fully subtractive algorithm 38
7 Maps on the real axis 41
7.1 Boole s transformation 41
7.2 Kemperman s examples 41
8 / expansions 43
8.1 / expansions of type A 43
8.2 / expansions of type B 44
8.3 Kakeya s theorem 44
9 Ergodicity 47
9.1 Measurable fibred systems 47
9.2 Ergodic fibred systems 49
9.3 Conditional expectation 50
9.4 Some examples 51
9.5 Renyi s theorem 54
10 Examples of non ergodic systems 59
10.1 Non ergodic fibred systems 59
10.2 Lehmer s cotangent algorithm 60
11 Distribution theorems 65
11.1 Oppenheim series 65
11.2 Some miscellaneous results 71
12 Invariant measures 75
12.1 Measure preserving maps 75
12.2 Conservative maps 76
12.3 Some examples of invariant measures 77
12.4 Some results for continued fractions 80
12.5 Infinite invariant measures 82
13 Kuzmin s equation 85
13.1 The transfer operator 85
13.2 Examples 87
CONTENTS xi
14 The Parry Daniels map 93
14.1 The Parry Daniels map 93
14.2 The case n = 3 96
15 Renyi s condition 105
15.1 Renyi s theorem 105
15.2 The folklore theorem 107
15.3 Examples 110
16 Auxiliary measures 111
16.1 Conjugate fibred systems 111
16.2 Auxiliary measures 113
17 First return map 119
17.1 First return map 119
17.2 Ergodic properties 124
17.3 Geodesic flows 127
18 Jump transformations I 129
1 9Q
18.1 Basic properties 129
18.2 Ergodic properties 131
18.3 Examples 133
18.4 Playback properties 135
18.5 Examples 137
19 Jump transformations II
19.1 Basic properties
19.2 Ergodic behaviour 144
20 Maps with indifferent fixed points 147
20.1 Indifferent fixed points 147
20.2 Invariant measures 148
20.3 Examples 154
21 Dual algorithms xo 157
21.1 Dual fibred systems 157
21.2 Natural extension 160
21.3 Examples 162
21.4 Japanese continued fractions 164
22 Piecewise fractional linear maps 171
22.1 Piecewise fractional linear maps 171
22.2 Invariant measures 172
xii CONTENTS
23 Multidimensional maps 179
23.1 Bran s algorithm 179
23.2 Selmer s algorithm 182
23.3 Jacobi Perron algorithm 183
23.4 Giiting s algorithm 186
23.5 Skew products 186
23.6 Complex continued fractions 187
24 Diophantine problems 191
24.1 Regular continued fractions 191
24.2 Diophantine problems I 196
24.3 Diophantine problems II 200
24.4 Singularization 205
25 The transfer operator 207
25.1 The transfer operator revisited 207
25.2 The Cesaro mean 210
26 The case of continued fractions 213
26.1 Historical remarks 213
26.2 Wirsing s theorem 214
26.3 Babenko s theorem 219
26.4 Remarks 225
27 The Lasota Yorke approach 227
27.1 Functions of bounded variation 227
27.2 Contracting lemma 229
27.3 Remarks 231
28 Distortion and other functionals 233
28.1 Distortion 233
28.2 Contracting lemma 234
29 More results on the Kuzmin operator 239
29.1 Kuzmin s approach 239
29.2 Miscellaneous results 240
29.3 5 unimodal maps 242
29.4 Infinite measures 244
CONTENTS xiii
30 The quadratic map 245
30.1 Appearance of fixed points 245
30.2 Ruelle s theorem 246
30.3 Pianigiani s theorem 250
30.4 Remarks 254
31 Exponents of convergence 255
31.1 Perron s identity 255
31.2 Ergodic theorems for matrices 258
A References 265
B Index 291
|
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| indexdate | 2024-07-09T17:47:07Z |
| institution | BVB |
| isbn | 0198534884 |
| language | English |
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| physical | XIII, 295 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
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| publisher | Clarendon Press [u.a.] |
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| spelling | Schweiger, Fritz 1942- Verfasser (DE-588)121576094 aut Ergodic theory of fibred systems and metric number theory Fritz Schweiger Oxford Clarendon Press [u.a.] 1995 XIII, 295 S. txt rdacontent n rdamedia nc rdacarrier Oxford science publications Dynamique différentiable ram Nombres, théorie des ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Number theory Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Faser Mathematik (DE-588)4153750-6 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 gnd rswk-swf Ergodentheorie (DE-588)4015246-7 s Faser Mathematik (DE-588)4153750-6 s Zahlentheorie (DE-588)4067277-3 s DE-604 Differenzierbares dynamisches System (DE-588)4137931-7 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006731290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schweiger, Fritz 1942- Ergodic theory of fibred systems and metric number theory Dynamique différentiable ram Nombres, théorie des ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Number theory Differenzierbares dynamisches System (DE-588)4137931-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Faser Mathematik (DE-588)4153750-6 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| subject_GND | (DE-588)4137931-7 (DE-588)4067277-3 (DE-588)4153750-6 (DE-588)4015246-7 |
| title | Ergodic theory of fibred systems and metric number theory |
| title_auth | Ergodic theory of fibred systems and metric number theory |
| title_exact_search | Ergodic theory of fibred systems and metric number theory |
| title_full | Ergodic theory of fibred systems and metric number theory Fritz Schweiger |
| title_fullStr | Ergodic theory of fibred systems and metric number theory Fritz Schweiger |
| title_full_unstemmed | Ergodic theory of fibred systems and metric number theory Fritz Schweiger |
| title_short | Ergodic theory of fibred systems and metric number theory |
| title_sort | ergodic theory of fibred systems and metric number theory |
| topic | Dynamique différentiable ram Nombres, théorie des ram Théorie ergodique ram Differentiable dynamical systems Ergodic theory Number theory Differenzierbares dynamisches System (DE-588)4137931-7 gnd Zahlentheorie (DE-588)4067277-3 gnd Faser Mathematik (DE-588)4153750-6 gnd Ergodentheorie (DE-588)4015246-7 gnd |
| topic_facet | Dynamique différentiable Nombres, théorie des Théorie ergodique Differentiable dynamical systems Ergodic theory Number theory Differenzierbares dynamisches System Zahlentheorie Faser Mathematik Ergodentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006731290&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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