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Walters P. — An introduction to ergodic theory

Walters P. — An introduction to ergodic theory

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Название: An introduction to ergodic theory

Автор: Walters P.

Аннотация:

This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1982

Количество страниц: 250

Добавлена в каталог: 01.04.2006

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Предметный указатель

$\alpha$ -limit set      123
$\beta$ -shift      178
$\omega$ -limit set      123
$\sigma$ -algebra      3
$\sigma$ -finite measure      34
Absolute continuity of measures      8
Affine transformation      21
Affine transformation, conditions for ergodicity      31
Affine transformation, conditions for minimality      122
Affine transformation, conditions for mixing      50
Affine transformation, conditions for topological transitivity      132
Affine transformation, conditions for unique ergodicity      162
Affine transformation, entropy      102 203
Affine transformation, non-wandering set      126
Affine transformation, topological entropy      203
Affinity of the entropy map      183
Algebra      3
Aperiodic non-negative matrix      16
Aperiodic transformation      97
Automorphism of a torus      15
Automorphism of a torus, conditions for ergodicity      31
Automorphism of a torus, conditions for expansiveness      143
Automorphism of a torus, conditions for mixing      50
Automorphism of a torus, conditions for topological transitivity      132
Automorphism of a torus, entropy      102
Automorphism of a torus, topological entropy      203
Axiom A diffeomorphism      130 226 229
Banach — Stone Theorem      136
Basis for a Hilbert space      10
Bernoulli automorphism      106
Bernoulli shift      105
Birkhoff's Ergodic Theorem      34
Block      5
Borel $\sigma$ -algebra of a topological space      10 18 146
Borel's theorem on normal numbers      35
Centre of a continuous transformation      126 157
Character theory      12
Choquet's theorem      153
Complete invariant      62 103
Completely positive entropy      110
Conditional entropy      80 84
Conditional expectation      9
Conjugacy of measure spaces      54
Conjugacy of measure-preserving transformations      59
Conjugacy, topological      133
Continued fractions      24
Continuous spectrum      48
Countable basis for a measure space      10 44
Countable Lebesgue spectrum      65
Countably additive      4
Density zero      43
Direct product of continuous transformations      174
Direct product of measure spaces      5
Direct product of measure-preserving transformations      23 99
Discrete spectrum for measure-preserving transformations      64 69
Discrete spectrum theorem      70
Discrete spectrum, topological      135
Distal      130
Dominated Convergence Theorem      8
Doob's martingale theorem      85
Eigenfunction of a continuous transformation      133
Eigenfunction of a measure-preserving transformation      48 68
Eigenvalue of a continuous transformation      133
Eigenvalue of a measure-preserving transformation      48 68
Endomorphism of a compact group      20
Endomorphism of a compact group, conditions for ergodicity      30
Endomorphism of a compact group, conditions for topological transitivity      132
Endomorphism of a compact group, conditions fur mixing      50
Endomorphism of a compact group, entropy      102 197
Endomorphism of a compact group, topological entropy      197
Endomorphism of a torus      15 20
Endomorphism of a torus, conditions for ergodicity      30 31
Endomorphism of a torus, conditions for mixing      50
Endomorphism of a torus, conditions for topological transitivity      132
Endomorphism of a torus, entropy      102 203
Endomorphism of a torus, topological entropy      203
Entropy map of a continuous transformation      182
Entropy of a Bernoulli shift      102
Entropy of a direct product      99
Entropy of a Markov shift      103
Entropy of a measure-preserving transformation      87
Entropy of a measure-preserving transformation w.r.t. a partition      86
Entropy of a partition      78
Entropy of a rotation      100 101
Entropy of an affine transformation      102 203
Entropy of an ergodic decomposition      186
Entropy, conditional      80 84
Entropy, sequence      114
Equilibrium state      224
Equivalent measures      8
Ergodic decomposition      33 153
Ergodic measure-preserving transformation      27
Ergodic theorem, Birkhoff      34
Ergodic theorem, maximal      37
Ergodic theorem, multiplicative      233
Ergodic theorem, subadditive      231
Ergodic theorem, von Neumann      36
Ergodicity conditions for affine transformations      31
Ergodicity conditions for automorphisms of compact groups      30
Ergodicity conditions for Markov-shifts      33 42
Ergodicity conditions for rotations      29 30
Exact endomorphism      115
Expansive homeomorphism      138 177 184 192 213 223 224
Factor of a continuous transformation      140
Factor of a measure-preserving transformation      61
Fatou's lemma      8
Finitary isomorphism      238
Finitely additive      4
Fourier series      13
Gauss measure      24
Generator for a measure-preserving transformation      96
Generator for an expansive homeomorphism      137
Generator, weak      137
Haar measure      11
Hilbert space      9
History of ergodic theory      1 239
Independent $\sigma$ -algebras      82
Independent partitions      82
Induced operator on $L^{p}$       25
Intrinsically ergodic      193
Invariant measure for a continuous transformation      151
Invariant of a measure-preserving transformation      62 66
Invertible measure-preserving transformation      19
Invertible mod 0      59
Irreducible non-negative matrix      16
Isomorphism of measure algebras      54
Isomorphism of measure spaces      53
Isomorphism of measure-preserving transformations      58
Jacobian of a measure-preserving transformation      116
Join of open covers      164
Join of partitions      76
Jordan decomposition theorem      3
Kingman's ergodic theorem      231
Kolmogorov automorphism      107
Kolmogorov automorphism, spectrum of      109
Kolmogorov — Sinai theorem      95
Lebesgue Covering Lemma      18
Lebesgue decomposition theorem      8

Lebesgue number of an open cover      18
Lebesgue space      55
Liapunov exponents      233 235
Markov measure      22
Markov shift      22
Markov shift, conditions for ergodicity      33 42
Markov shift, conditions for mixing      51
Markov shift, conditions to be a Kolmogorov automorphism      111
Markov shift, entropy      103
Maximal ergodic theorem      37
McMillan's theorem      93
Measure      3
Measure of maximal entropy      191
Measure, $\sigma$ -finite      34
Measure, algebra      54
Measure, describing the distribution of periodic points      204
Measure, Haar      11
Measure, Markov      22
Measure, Parry      194
Measure, product      5
Measure, regular      11 147
Measure, space      3
Measure-preserving transformation      19
Measureable space      3
Measureable transformation      19
Minimal set      121
Minimal transformation      120
Mixing, strong      40
Mixing, weak      40
Monotone class      6
Monotone Convergence Theorem      7
Multiplicative ergodic theorem      233
Nilmanifold      112
Nilmanifold, automorphism      112
Non-atomic measure      48
Non-singular transformation      237
Non-wandering set      124 156
Normal number      35
North-south map      120
Orbit      120
Orbit equivalence      237
Ornstein's theorem      105
Orthogonal complement      10
Oseledets' ergodic theorem      233
Parry measure      194
Partition      75
Partition, entropy of      78
Partition, join of      76
Partition, refinement of      76
Periodic point      122 129 130 144 157
Perron — Frobenius theory      16
Pinsker $\sigma$ -algebra      113
Poincare's reccurrence theorem      26
Point measure      146
Pressure      208 209 210
Probability measure      3
Probability vector      5 21
Product, $\sigma$ -algebra      5
Product, measure      5
Product, transformation      23
Pure-point spectrum of a continuous transformation      135
Pure-point spectrum of a measure-preserving transformation      69
Quasi-invariant measure      237
Radon — Nikodym derivative      8
Radon — Nikodym theorem      8
rectangle      5
Recurrence theorem of Poincare      26
Refinement of open covers      164
Refinement of partitions      76
Regular measure      11 147
Renewal Theorem      17
Riesz representation theorem      148
Root of a measure-preserving transformation      106
Rotation      20 72
Rotation, conditions for ergodicity      29 30
Rotation, conditions for minimality      122
Rotation, conditions for mixing      50
Rotation, conditions for topological transitivity      132
Rotation, conditions for unique ergodicity      162
Rotation, entropy      100 101
Rotation, non-wandering set      126 157
Rotation, pressure      221
Rotation, topological entropy      170 179
Schwarz inequality      10
Semi-algebra      3
Semi-conjugate image      61
Semi-simple homeomorphism      121
Separated set      169
Sequence entropy      114
Shannon — McMillan — Brieman theorem      93
Shift, $\beta$       178
Shift, Markov      22
Shift, one-sided ( $p_{0},...,p_{k-1}$ )      21
Shift, topological      119
Shift, two-sided ( $p_{0},...,p_{k-1}$ )      21
Signed measure      3
Simple function      7
Singular measures      8
Spanning set      168
Spectral invariant      66
Spectral isomorphism      63
Spectral theorem for unitary operators      48
Spectrum, continuous      48
Spectrum, discrete      64
Spectrum, Lebesgue      65
State space      21 105
Stochastic matrix      22
Stone — Weierstrass theorem      135
Strong mixing      40
Subadditive ergodic theorem      231
Subshift of finite type      119
Tangent functional      224
Time average      35
Topological conjugacy      133
Topological entropy      166 169
Topological entropy of $\beta$ -shift      178
Topological entropy of affine transformations      197 203
Topological entropy of differentiable maps      181
Topological entropy of homeomorphisms of the circle      179
Topological entropy of homeomorphisms of the interval      180
Topological entropy of linear transformations      201
Topological entropy of rotations      170 179
Topological entropy of shift      177 178
Topological Markov chain      119
Topological pressure      208 209 210
Topological pressure of a rotation      221
Topological transitivity      127
Topological transitivity, one-sided      127
Transformations of intervals      238
Uniformly equivalent metrics      171
Unique measure with maximal entropy      193
Uniquely ergodic      158
Unitary operator      10
Variational Principle      188 218
von Neumann ergodic theorem      36
Wandering point      124
Weak mixing      40
weak*-topology      148

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