Dynamical zeta functions, Nielsen theory and Reidemeister torsion:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2000
|
| Series: | American Mathematical Society: Memoirs of the American Mathematical Society
699 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Volume 147, Number 699 (third of 4 numbers) Zugl.: Greifswald, Univ., Habil.-Schr., 1998 |
| Physical Description: | XI, 146 S. |
| ISBN: | 0821820907 |
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MARC
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| 245 | 1 | 0 | |a Dynamical zeta functions, Nielsen theory and Reidemeister torsion |c Alexander Fel'shtyn |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2000 | |
| 300 | |a XI, 146 S. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a American Mathematical Society: Memoirs of the American Mathematical Society |v 699 | |
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| 500 | |a Zugl.: Greifswald, Univ., Habil.-Schr., 1998 | ||
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Record in the Search Index
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| adam_text |
Contents
Introduction 1
0.1 From Riemann zeta function to dynamical zeta functions . 1
0.1.1 Riemann zeta function 1
0.1.2 Problems concerning zeta functions 2
0.1.3 Important types of zeta functions 3
0.1.4 Hasse Weil zeta function 3
0.1.5 Dynamical zeta functions 4
0.2 Dynamical zeta functions and Nielsen fixed point theory . 6
0.3 Congruences for Reidemeister numbers 8
0.4 Reidemeister torsion 9
0.5 Table of contents 11
1 Nielsen Fixed Point Theory 14
1.1 History 14
1.2 Lifting classes and fixed point classes 17
1.2.1 The influence of a homotopy 18
1.3 Reidemeister numbers 19
1.3.1 Reidemeister numbers of a continuous map 19
1.3.2 Reidemeister numbers of a group endomorphism . 21
1.4 Nielsen numbers of a continuous map 22
1.4.1 The fixed point index 22
1.4.2 Nielsen numbers 25
1.4.3 The least number of fixed points 26
2 The Reidemeister zeta function 27
2.1 A Convolution Product 27
2.2 Pontryagin Duality 30
vii
viii CONTENTS
2.3 Eventually commutative endomorphisms 32
2.3.1 Trace formula for the Reidemeister numbers of eventu¬
ally commutative endomorphisms 32
2.3.2 Rationality of Reidemeister zeta functions of eventu¬
ally commutative endomorphisms first proof 35
2.3.3 Functional equation for the Reidemeister zeta function
of an eventually commutative endomorphism 39
2.3.4 Rationality of Reidemeister zeta functions of eventu¬
ally commutative endomorphisms second proof. . 41
2.3.5 Connection of the Reidemeister zeta function with the
Lefschetz zeta function of the dual map 42
2.4 Endomorphisms of finite groups 44
2.5 Endomorphisms of the direct sum of a free Abelian and a
finite group 48
2.6 Endomorphisms of nilpotent groups 53
2.6.1 Functional equation 54
2.7 The Reidemeister zeta function and group extensions 55
2.8 The Reidemeister zeta function of a continuous map 56
2.8.1 The Reidemeister zeta function of a continuous map
and Serre bundles 59
3 The Nielsen zeta function 60
3.1 Radius of Convergence of the Nielsen zeta function 60
3.1.1 Topological entropy 60
3.1.2 Algebraic lower estimation for the Radius of Convergence 62
3.2 Nielsen zeta function of a periodic map 66
3.3 Orientation preserving homeomorphisms of a compact surface 69
3.3.1 Geometry of the Mapping Torus and Radius of Con¬
vergence 70
3.4 The Jiang subgroup and the Nielsen zeta function 72
3.5 Polyhedra with finite fundamental group 76
3.6 Nielsen zeta function in other special cases 82
3.6.1 Pseudo Anosov homeomorphism of a compact surface 82
3.7 The Nielsen zeta function and Serre bundles 83
3.8 Examples 85
CONTENTS ix
4 Reidemeister and Nielsen zeta functions modulo normal sub¬
group, minimal dynamical zeta functions 88
4.1 Reidemeister and Nielsen zeta functions modulo a normal sub¬
group 88
4.1.1 Radius of Convergence of the mod K Nielsen zeta func¬
tion 89
4.1.2 mod K Nielsen zeta function of a periodic map 90
4.2 Minimal dynamical zeta function 92
4.2.1 Radius of Convergence of the minimal zeta function . . 92
5 Congruences for Reidemeister and Nielsen numbers 94
5.1 Irreducible Representation and the Unitary Dual of G 94
5.2 Endomorphism of the Direct Sum of a Free Abelian and a
Finite Group 95
5.3 Endomorphism of almost Abelian
groups 97
5.3.1 Compact Groups 98
5.3.2 Almost Abelian groups 99
5.4 Endomorphisms of nilpotent groups 101
5.5 Main Theorem 102
5.6 Congruences for Reidemeister numbers of a continuous map . 104
5.7 Congruences for Reidemeister numbers of equivariant group
endomorphisms 105
5.8 Congruences for Reidemeister numbers of equivariant maps . . 109
5.9 Congruences for Nielsen numbers of a continuous map 110
5.10 Some conjectures for wider classes of
groups 112
6 The Reidemeister torsion 113
6.1 Preliminaries 113
6.2 The Reidemeister zeta Function and the Reidemeister Torsion
of the Mapping Torus of the dual map 117
6.3 The connection between the Reidemeister torsion, eta invariant,
the Rochlin invariant and theta multipliers via the dynamical
zeta functions 119
6.3.1 Rochlin invariant 119
6.3.2 The Bismut Freed Witten Holonomy Theorem 120
x CONTENTS
6.3.3 Connection with Reiderneister torsion 123
6.3.4 The Reidemeister torsion and theta multipliers 124
6.4 Topology of an attraction domain and the Reidemeister torsion 125
6.4.1 Introduction 125
6.4.2 Morse Smale systems 126
6.4.3 A formula for the Euler characteristic 130
6.4.4 The Reidemeister torsion of the level surface of a Lya
punov function and of the attraction domain of the
attractor 131
6.5 Integrable Hamiltonian systems and the Reidemeister torsion . 135 |
| any_adam_object | 1 |
| author | Fel'štyn, Aleksandr L. 1952- |
| author_GND | (DE-588)120358093 |
| author_facet | Fel'štyn, Aleksandr L. 1952- |
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| classification_rvk | SK 300 SK 520 SI 130 |
| classification_tum | MAT 300d |
| ctrlnum | (OCoLC)614878152 (DE-599)BVBBV013363976 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013363976 |
| illustrated | Not Illustrated |
| indexdate | 2025-03-03T13:02:18Z |
| institution | BVB |
| isbn | 0821820907 |
| language | English |
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| physical | XI, 146 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
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| publisher | American Math. Soc. |
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| series | American Mathematical Society: Memoirs of the American Mathematical Society |
| series2 | American Mathematical Society: Memoirs of the American Mathematical Society |
| spelling | Fel'štyn, Aleksandr L. 1952- Verfasser (DE-588)120358093 aut Dynamical zeta functions, Nielsen theory and Reidemeister torsion Alexander Fel'shtyn Providence, RI American Math. Soc. 2000 XI, 146 S. txt rdacontent n rdamedia nc rdacarrier American Mathematical Society: Memoirs of the American Mathematical Society 699 Volume 147, Number 699 (third of 4 numbers) Zugl.: Greifswald, Univ., Habil.-Schr., 1998 Fixpunkttheorie (DE-588)4293945-8 gnd rswk-swf Zetafunktion (DE-588)4190764-4 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Zetafunktion (DE-588)4190764-4 s Fixpunkttheorie (DE-588)4293945-8 s DE-604 American Mathematical Society: Memoirs of the American Mathematical Society 699 (DE-604)BV008000141 699 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009115365&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fel'štyn, Aleksandr L. 1952- Dynamical zeta functions, Nielsen theory and Reidemeister torsion American Mathematical Society: Memoirs of the American Mathematical Society Fixpunkttheorie (DE-588)4293945-8 gnd Zetafunktion (DE-588)4190764-4 gnd |
| subject_GND | (DE-588)4293945-8 (DE-588)4190764-4 (DE-588)4113937-9 |
| title | Dynamical zeta functions, Nielsen theory and Reidemeister torsion |
| title_auth | Dynamical zeta functions, Nielsen theory and Reidemeister torsion |
| title_exact_search | Dynamical zeta functions, Nielsen theory and Reidemeister torsion |
| title_full | Dynamical zeta functions, Nielsen theory and Reidemeister torsion Alexander Fel'shtyn |
| title_fullStr | Dynamical zeta functions, Nielsen theory and Reidemeister torsion Alexander Fel'shtyn |
| title_full_unstemmed | Dynamical zeta functions, Nielsen theory and Reidemeister torsion Alexander Fel'shtyn |
| title_short | Dynamical zeta functions, Nielsen theory and Reidemeister torsion |
| title_sort | dynamical zeta functions nielsen theory and reidemeister torsion |
| topic | Fixpunkttheorie (DE-588)4293945-8 gnd Zetafunktion (DE-588)4190764-4 gnd |
| topic_facet | Fixpunkttheorie Zetafunktion Hochschulschrift |
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