Dynamical systems and processes:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Society
2009
|
| Series: | IRMA lectures in mathematics and theoretical physics
14 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 761 S. |
| ISBN: | 9783037190463 |
Staff View
MARC
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| 100 | 1 | |a Weber, Michel |d 1949- |e Verfasser |0 (DE-588)139154698 |4 aut | |
| 245 | 1 | 0 | |a Dynamical systems and processes |c Michel Weber |
| 264 | 1 | |a Zürich |b European Mathematical Society |c 2009 | |
| 300 | |a XII, 761 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a IRMA lectures in mathematics and theoretical physics |v 14 | |
| 650 | 4 | |a Convergence | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Dynamics | |
| 650 | 4 | |a Ergodic theory | |
| 650 | 4 | |a Probabilities | |
| 650 | 4 | |a Spectral theory (Mathematics) | |
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Record in the Search Index
| _version_ | 1804137959195148288 |
|---|---|
| adam_text | Contents
Preface
v
Part I Spectral theorems and convergence in mean
1
1
The
von
Neumann theorem and spectral regularization
3
1.1
Bochner-Herglotz lemma
.......................... 3
1.2
The spectral inequality
........................... 8
1.3
The
von
Neumann theorem
......................... 10
1.4
The spectral regularization inequality
.................... 26
1.5
Moving averages
.............................. 44
1.6
Uniform distribution mod a
-
the Weyl criterion
.............. 51
1.7
The van
der Corput
principle
........................ 55
2
Spectral representation of weakly stationary processes
61
2.1
Weakly stationary processes
........................ 61
2.2
Spectral representation of unitary operators
................ 64
2.3
Elements of stochastic integration
..................... 76
2.4
Spectral representation of weakly stationary processes
........... 78
2.5
Weakly stationary sequences and orthogonal series
............ 80
2.6
Gaposhkin s spectral criterion
....................... 85
Part II Ergodic Theorems
91
3
Dynamical systems
-
ergodicity and mixing
93
3.1
Measurable dynamical systems
-
topological dynamical systems
..... 93
3.2
Ergodicity of a dynamical system
...................... 101
3.3
Weak mixing, strong mixing, continuous spectrum
............. 103
3.4
Spectral mixing theorem
.......................... 110
3.5
Other equivalences and other forms of mixing
............... 114
3.6
Examples
.................................. 121
4
Pointwise ergodic theorems
129
4.1
Birkhoff s pointwise theorem
........................ 129
4.2
Dominated ergodic theorems
........................ 139
4.3
Classes Llog 1
L
.............................. 144
4.4
A converse
.................................. 145
χ
Contents
4.5
Speed of
convergence
............................ 148
4.6
Oscillation functions of ergodic averages
.................. 152
4.7
Wiener-Wintner theorem
.......................... 165
4.8
Weighted ergodic averages
......................... 168
4.9
Subsequence averages
............................ 193
5
Banach principle and continuity principle
200
5.1
Banach principle
.............................. 200
5.2
Continuity principle
............................. 206
5.3
Applications
................................. 217
5.4
A principle of domination
—
conjugacy lemma
............... 226
6
Maximal operators and Gaussian processes
230
6.1
Some liaison theorems
........................... 230
6.2
Two preliminary lemmas
.......................... 242
6.3
Proof of Theorem
6.1.1........................... 247
6.4
Proof of Theorem
6.1.6........................... 249
6.5
The case Lp,
1 <
ρ
< 2 .......................... 254
6.6
A remarkable
GB set
property
....................... 259
7
The central limit theorem for dynamical systems
267
7.1
Introduction and preliminaries
....................... 267
7.2
A theorem of Burton and
Denker...................... 269
7.3
The central limit theorem for orbits
..................... 284
7.4
A theorem of
Volný
............................. 289
7.5
CLT for rotations
.............................. 291
7.6
Lacunary series and convergence in variation
............... 315
Part
ΠΙ
Methods arising from the theory of stochastic processes
339
8
The metric entropy method
341
8.1
Introduction and general results
....................... 341
8.2
A theorem of Stechkin
........................... 349
8.3
An application to the quantitative
Borei-Cantelli
lemma
.......... 353
8.4
Application to
Gál-Koksma s
theorems
.................. 364
8.5
An application to the supremum of random polynomials
.......... 369
8.6
Application to a.s. convergence of weighted series of contractions
.... 387
8.7
An application to random perturbation of intersective sets
......... 403
8.8
An application to the discrepancy of some random sequences
....... 409
8.9
An application to random Dirichlet polynomials
.............. 415
9
The majorizing measure method
433
9.1
Introduction
-
the exponential case
..................... 433
Contents xi
9.2
A general approach
............................. 438
9.3
A useful criterion
.............................. 447
9.4
Proof of Theorem
9.3.3........................... 457
9.5
Proof of Theorems
9.3.10
and
9.3.11.................... 469
9.6
Proof of Theorem
9.3.12
and some examples
............... 471
9.7
A stronger form of Salem-Zygmund s estimate
.............. 475
9.8
Some examples and discussion
....................... 478
9.9
Uniform convergence of random Fourier series
.............. 488
10
Gaussian processes
491
10.1
Gaussian variables and correlation estimates
................ 491
10.2 0-1
laws, integrability and comparison lemmas
.............. 504
10.3
Regularity and irregularity of Gaussian processes
............. 510
10.4
Gaussian
suprema
.............................. 517
10.5
Oscillations of Gaussian Stem s elements
................. 529
10.6
Tightness of Gaussian Stein s elements
................... 537
Part IV Three studies
547
11
Riemannsums
549
11.1
Introduction
................................. 549
11.2
The results of
Jessen
and
Rudin
....................... 551
11.3
Individual theorems of spectral type
.................... 554
11.4
Breadth and dimension
........................... 557
11.5
Bourgain s results
.............................. 562
11.6
Connection with number theory
...................... 565
11.7
Riemann sums and the randomly sampled trigonometric system
..... 573
11.8
Almost sure convergence and square functions of Riemann sums
..... 587
12
A study of the system (f(nx))
601
12.1
Introduction and mean convergence
.................... 601
12.2
Almost sure convergence
-
sufficient conditions
.............. 611
12.3
Almost sure convergence
-
necessary conditions
.............. 634
12.4
Random sequences
............................. 642
13
Divisors and random walks
659
13.1
Introduction
................................. 659
13.2
Value distribution and small divisors of Bernoulli sums
.......... 661
13.3
An
LIL
for arithmetic functions
....................... 675
13.4
On the order of magnitude of the divisor functions
............. 685
13.5
Value distribution of the divisors of n2
+ 1................. 691
13.6
Value distribution of the divisors of Rademacher sums
........... 699
13.7
The functional equation and the
Lindelof
Hypothesis
........... 701
xii Contents
13.8 An extxemal
divisor
case
..........................711
Bibliography
729
Index 759
|
| adam_txt |
Contents
Preface
v
Part I Spectral theorems and convergence in mean
1
1
The
von
Neumann theorem and spectral regularization
3
1.1
Bochner-Herglotz lemma
. 3
1.2
The spectral inequality
. 8
1.3
The
von
Neumann theorem
. 10
1.4
The spectral regularization inequality
. 26
1.5
Moving averages
. 44
1.6
Uniform distribution mod a
-
the Weyl criterion
. 51
1.7
The van
der Corput
principle
. 55
2
Spectral representation of weakly stationary processes
61
2.1
Weakly stationary processes
. 61
2.2
Spectral representation of unitary operators
. 64
2.3
Elements of stochastic integration
. 76
2.4
Spectral representation of weakly stationary processes
. 78
2.5
Weakly stationary sequences and orthogonal series
. 80
2.6
Gaposhkin's spectral criterion
. 85
Part II Ergodic Theorems
91
3
Dynamical systems
-
ergodicity and mixing
93
3.1
Measurable dynamical systems
-
topological dynamical systems
. 93
3.2
Ergodicity of a dynamical system
. 101
3.3
Weak mixing, strong mixing, continuous spectrum
. 103
3.4
Spectral mixing theorem
. 110
3.5
Other equivalences and other forms of mixing
. 114
3.6
Examples
. 121
4
Pointwise ergodic theorems
129
4.1
Birkhoff's pointwise theorem
. 129
4.2
Dominated ergodic theorems
. 139
4.3
Classes Llog"1
L
. 144
4.4
A converse
. 145
χ
Contents
4.5
Speed of
convergence
. 148
4.6
Oscillation functions of ergodic averages
. 152
4.7
Wiener-Wintner theorem
. 165
4.8
Weighted ergodic averages
. 168
4.9
Subsequence averages
. 193
5
Banach principle and continuity principle
200
5.1
Banach principle
. 200
5.2
Continuity principle
. 206
5.3
Applications
. 217
5.4
A principle of domination
—
conjugacy lemma
. 226
6
Maximal operators and Gaussian processes
230
6.1
Some liaison theorems
. 230
6.2
Two preliminary lemmas
. 242
6.3
Proof of Theorem
6.1.1. 247
6.4
Proof of Theorem
6.1.6. 249
6.5
The case Lp,
1 <
ρ
< 2 . 254
6.6
A remarkable
GB set
property
. 259
7
The central limit theorem for dynamical systems
267
7.1
Introduction and preliminaries
. 267
7.2
A theorem of Burton and
Denker. 269
7.3
The central limit theorem for orbits
. 284
7.4
A theorem of
Volný
. 289
7.5
CLT for rotations
. 291
7.6
Lacunary series and convergence in variation
. 315
Part
ΠΙ
Methods arising from the theory of stochastic processes
339
8
The metric entropy method
341
8.1
Introduction and general results
. 341
8.2
A theorem of Stechkin
. 349
8.3
An application to the quantitative
Borei-Cantelli
lemma
. 353
8.4
Application to
Gál-Koksma's
theorems
. 364
8.5
An application to the supremum of random polynomials
. 369
8.6
Application to a.s. convergence of weighted series of contractions
. 387
8.7
An application to random perturbation of intersective sets
. 403
8.8
An application to the discrepancy of some random sequences
. 409
8.9
An application to random Dirichlet polynomials
. 415
9
The majorizing measure method
433
9.1
Introduction
-
the exponential case
. 433
Contents xi
9.2
A general approach
. 438
9.3
A useful criterion
. 447
9.4
Proof of Theorem
9.3.3. 457
9.5
Proof of Theorems
9.3.10
and
9.3.11. 469
9.6
Proof of Theorem
9.3.12
and some examples
. 471
9.7
A stronger form of Salem-Zygmund's estimate
. 475
9.8
Some examples and discussion
. 478
9.9
Uniform convergence of random Fourier series
. 488
10
Gaussian processes
491
10.1
Gaussian variables and correlation estimates
. 491
10.2 0-1
laws, integrability and comparison lemmas
. 504
10.3
Regularity and irregularity of Gaussian processes
. 510
10.4
Gaussian
suprema
. 517
10.5
Oscillations of Gaussian Stem's elements
. 529
10.6
Tightness of Gaussian Stein's elements
. 537
Part IV Three studies
547
11
Riemannsums
549
11.1
Introduction
. 549
11.2
The results of
Jessen
and
Rudin
. 551
11.3
Individual theorems of spectral type
. 554
11.4
Breadth and dimension
. 557
11.5
Bourgain's results
. 562
11.6
Connection with number theory
. 565
11.7
Riemann sums and the randomly sampled trigonometric system
. 573
11.8
Almost sure convergence and square functions of Riemann sums
. 587
12
A study of the system (f(nx))
601
12.1
Introduction and mean convergence
. 601
12.2
Almost sure convergence
-
sufficient conditions
. 611
12.3
Almost sure convergence
-
necessary conditions
. 634
12.4
Random sequences
. 642
13
Divisors and random walks
659
13.1
Introduction
. 659
13.2
Value distribution and small divisors of Bernoulli sums
. 661
13.3
An
LIL
for arithmetic functions
. 675
13.4
On the order of magnitude of the divisor functions
. 685
13.5
Value distribution of the divisors of n2
+ 1. 691
13.6
Value distribution of the divisors of Rademacher sums
. 699
13.7
The functional equation and the
Lindelof
Hypothesis
. 701
xii Contents
13.8 An extxemal
divisor
case
.711
Bibliography
729
Index 759 |
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| id | DE-604.BV035029303 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:49:14Z |
| indexdate | 2024-07-09T21:20:34Z |
| institution | BVB |
| isbn | 9783037190463 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016698306 |
| oclc_num | 436628341 |
| open_access_boolean | |
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| owner_facet | DE-355 DE-BY-UBR DE-384 DE-29T DE-20 DE-824 DE-11 DE-83 |
| physical | XII, 761 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | European Mathematical Society |
| record_format | marc |
| series | IRMA lectures in mathematics and theoretical physics |
| series2 | IRMA lectures in mathematics and theoretical physics |
| spelling | Weber, Michel 1949- Verfasser (DE-588)139154698 aut Dynamical systems and processes Michel Weber Zürich European Mathematical Society 2009 XII, 761 S. txt rdacontent n rdamedia nc rdacarrier IRMA lectures in mathematics and theoretical physics 14 Convergence Differentiable dynamical systems Dynamics Ergodic theory Probabilities Spectral theory (Mathematics) Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s DE-604 IRMA lectures in mathematics and theoretical physics 14 (DE-604)BV014300756 14 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Weber, Michel 1949- Dynamical systems and processes IRMA lectures in mathematics and theoretical physics Convergence Differentiable dynamical systems Dynamics Ergodic theory Probabilities Spectral theory (Mathematics) Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4013396-5 |
| title | Dynamical systems and processes |
| title_auth | Dynamical systems and processes |
| title_exact_search | Dynamical systems and processes |
| title_exact_search_txtP | Dynamical systems and processes |
| title_full | Dynamical systems and processes Michel Weber |
| title_fullStr | Dynamical systems and processes Michel Weber |
| title_full_unstemmed | Dynamical systems and processes Michel Weber |
| title_short | Dynamical systems and processes |
| title_sort | dynamical systems and processes |
| topic | Convergence Differentiable dynamical systems Dynamics Ergodic theory Probabilities Spectral theory (Mathematics) Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Convergence Differentiable dynamical systems Dynamics Ergodic theory Probabilities Spectral theory (Mathematics) Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016698306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV014300756 |
| work_keys_str_mv | AT webermichel dynamicalsystemsandprocesses |