Cohomological theory of dynamical Zeta functions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2001
|
| Series: | Progress in mathematics
194 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009177476&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | X, 709 S. graph. Darst. |
| ISBN: | 376436405X |
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| 100 | 1 | |a Juhl, Andreas |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Cohomological theory of dynamical Zeta functions |c Andreas Juhl |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2001 | |
| 300 | |a X, 709 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Progress in mathematics |v 194 | |
| 650 | 4 | |a Functions, Zeta | |
| 650 | 4 | |a Homology theory | |
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Record in the Search Index
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Contents
Preface ix
Chapter 1. Introduction 1
1.1. The dynamical zeta functions 1
1.2. The motivations of the cohomological theory 4
1.2.1. Quantization of chaos 4
1.2.2. Uniform descriptions of the divisors of zeta functions 7
1.3. The contents of the book 13
1.3.1. Spectral theory on X, Lefschetz formulas on SX and
F invariant distributions on the ideal boundary Sn~l 13
1.3.2. Harmonic currents and divisors of the zeta functions.
The main ideas 25
1.3.3. Harmonic currents and divisors of the zeta functions.
The results and the conjectures 30
Chapter 2. Preliminaries 63
2.1. General notation 63
2.2. Lie theory related to the conformal group 63
2.3. Hyperbolic spaces as Riemannian manifolds
and symmetric spaces 67
2.4. n~ homology, n~ cohomology and Osborne's character formula 75
2.5. Induced representations and differential intertwining operators . 76
2.6. The classification of the unitary irreducible representations
of the Lorentz group SO(l,n)° 79
Chapter 3. Zeta Functions of the Geodesic Flow
of Compact Locally Symmetric Manifolds 87
3.1. Spectral theory of operators 88
3.2. The dynamical Lefschetz formula 103
3.3. Explicit formulas for the divisor in terms of
complexes on the ideal boundary 177
3.4. Patterson's conjecture 218
Chapter 4. Operators and Complexes 231
4.1. Equivariant differential operators and equivariant differential
complexes for the twisted geodesic flows 231
4.1.1. The de Rham complexes and the canonical complexes 231
vi Contents
4.1.2. Geometry of the operators d~,6~,D+ and A+ 262
4.1.2.1. The operator Dp and the spaces CC{XP'O)(SY, Va) 264
4.1.2.2. The complexes on CT^p(SY, Va) 276
4.1.2.3. The Euler operator 281
4.1.2.4. More commutator relations 283
4.1.2.5. The operators D+ and DCT 288
4.1.2.6. The operators 6~(fl~A) and fl~ A 8~ 290
4.1.2.7. The spaces S{^0) (SY, Va) 292
4.2. The Bruhat and Iwasawa models 296
4.2.1. The Bruhat models of the operators D+ and ? 297
4.2.2. The Iwasawa models of the operators D+ and ? 315
Chapter 5. The Verma Complexes on SY and SX 331
5.1. The Bruhat models of the Verma complexes on SY 331
5.2. The Iwasawa models of the Verma complexes on SY 343
5.3. The Verma complexes on SX 360
Chapter 6. Harmonic Currents and Canonical Complexes 373
6.1. Equivariant Hodge decomposition of CC^J_%(SY) for A 0 No • • • ¦ 374
6.2. Equivariant right parametrices of D+ and 6~ for A £ —No 392
6.3. Hodge decomposition of CC(Ap 0) (SX) for A £ No 410
6.4. Hodge decomposition of CC^p 0) (SX) for A € No 412
6.5. The system \3~uj = 0, Do; = 0 and exotic currents 432
6.6. The functional equation as an index formula 452
Chapter 7. Divisors and Harmonic Currents 469
7.1. The divisor of the Selberg zeta function 469
7.2. The divisor of the Ruelle zeta function 480
7.3. Harmonic currents which are constant on the leaves of J*~ 486
7.4. The Ruelle zeta functions of the geodesic flow of T\H4 493
Chapter 8. Further Developments and Open Problems 519
8.1. The divisor of Zs for convex cocompact groups 519
8.1.1. Scattering operators, extension operators and invariant
currents on the limit set 525
8.1.2. F cohomology of holomorphic families of hyperfunctions
on the limit set 569
8.1.3. The embedded case 585
8.1.4. F cohomology and harmonic currents 607
8.2. Miscellaneous problems and comments 624
Contents vii
8.2.1. The relations between the various definitions of twisted
Selberg zeta functions 624
8.2.2. Dynamical theta functions 625
8.2.3. Zeta functions and zeta regularized determinants 637
8.2.4. Closed ranges in the tangential complex of
the stable foliation 639
8.2.5. The spaces CV{xn_l0)(SX) and the operators H0{s) 640
8.2.6. Hodge decompositions 644
8.2.7. The equation d~u = 6 in the twisted case 647
8.2.8. Patterson's conjecture as a fixed point formula 650
8.2.9. Topological singularities and group cohomology 652
8.2.10. Meromorphic extension of Selberg zeta functions
and smoothness of T* 655
8.2.11. Zeta functions of the geodesic flow of rank one spaces 656
8.2.12. Lefschetz formulas and zeta functions for flows associated
to locally symmetric spaces of higher rank 657
8.2.13. Zeta functions for negative curvature spaces 661
8.2.14. Lefschetz fixed point formulas for foliations 666
8.3. Some historical comments 667
Chapter 9. A Summary of Important Formulas 673
Bibliography 687
Index of Equations 703
Index 707 |
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| illustrated | Illustrated |
| indexdate | 2025-03-03T13:02:18Z |
| institution | BVB |
| isbn | 376436405X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009177476 |
| oclc_num | 45223589 |
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| owner_facet | DE-29T DE-355 DE-BY-UBR DE-824 DE-11 DE-188 |
| physical | X, 709 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Juhl, Andreas Verfasser aut Cohomological theory of dynamical Zeta functions Andreas Juhl Basel [u.a.] Birkhäuser 2001 X, 709 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 194 Functions, Zeta Homology theory Kohomologietheorie (DE-588)4164610-1 gnd rswk-swf Zetafunktion (DE-588)4190764-4 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s Kohomologietheorie (DE-588)4164610-1 s DE-604 Progress in mathematics 194 (DE-604)BV000004120 194 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009177476&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Juhl, Andreas Cohomological theory of dynamical Zeta functions Progress in mathematics Functions, Zeta Homology theory Kohomologietheorie (DE-588)4164610-1 gnd Zetafunktion (DE-588)4190764-4 gnd |
| subject_GND | (DE-588)4164610-1 (DE-588)4190764-4 |
| title | Cohomological theory of dynamical Zeta functions |
| title_auth | Cohomological theory of dynamical Zeta functions |
| title_exact_search | Cohomological theory of dynamical Zeta functions |
| title_full | Cohomological theory of dynamical Zeta functions Andreas Juhl |
| title_fullStr | Cohomological theory of dynamical Zeta functions Andreas Juhl |
| title_full_unstemmed | Cohomological theory of dynamical Zeta functions Andreas Juhl |
| title_short | Cohomological theory of dynamical Zeta functions |
| title_sort | cohomological theory of dynamical zeta functions |
| topic | Functions, Zeta Homology theory Kohomologietheorie (DE-588)4164610-1 gnd Zetafunktion (DE-588)4190764-4 gnd |
| topic_facet | Functions, Zeta Homology theory Kohomologietheorie Zetafunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009177476&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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