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Ergodic Theorems for Group Actions : Informational and Thermodynamical Aspects - Arkady Tempelman

Ergodic Theorems for Group Actions

Informational and Thermodynamical Aspects


Published: 1st July 1992
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This volume is devoted to generalizations of the classical Birkhoff and von Neuman ergodic theorems to semigroup representations in Banach spaces, semigroup actions in measure spaces, homogeneous random fields and random measures on homogeneous spaces. The ergodicity, mixing and quasimixing of semigroup actions and homogeneous random fields are considered as well. In particular homogeneous spaces, on which all homogeneous random fields are quasimixing are introduced and studied (the n-dimensional Euclidean and Lobachevsky spaces with n>=2, and all simple Lie groups with finite centre are examples of such spaces. Also dealt with are applications of general ergodic theorems for the construction of specific informational and thermodynamical characteristics of homogeneous random fields on amenable groups and for proving general versions of the McMillan, Breiman and Lee-Yang theorems. A variational principle which characterizes the Gibbsian homogeneous random fields in terms of the specific free energy is also proved. The book has eight chapters, a number of appendices and a substantial list of references. For researchers whose works involves probability theory, ergodic theory, harmonic analysis, measure theory and statistical Physics.

Series Editor's Preface
Introductionp. 1
Basic ergodic and mean-value theoremsp. 1
Further ergodic theorems for operatorsp. 5
Means, averaging sequences and ergodic theorems on groupsp. 6
Means and averageable functionsp. 10
Means and quasi-averageable functionsp. 10
Averageable functionsp. 14
Averageable elements in general Banach spacesp. 24
Fixed point theoremsp. 26
Some classes of averageable functions and Banach space elementsp. 32
Means of positive definite functionsp. 42
Averageability of homogeneous random fields and of orbital functions of dynamical systemsp. 44
Ergodicity and mixingp. 51
Main definitionsp. 51
Ergodicity and mixing of Gaussian homogeneous random functionsp. 62
WM-spacesp. 68
SM-spacesp. 76
Averaging sequences. Universal ergodic theoremsp. 85
General mean value theoremsp. 85
Averaging nets on group extensions and productsp. 88
Universal ergodic theorems on amenable semigroupsp. 95
Existence of universally averaging nets of measures on general groupsp. 98
Universally O[subscript W(B)]-averaging sequencesp. 103
Averaging with respect to contractive representationsp. 111
C[subscript a]-averaging netsp. 111
Mean ergodic theoremsp. 117
Mean averaging nets: definition and existencep. 117
Mean ergodic theorems on amenable semigroupsp. 127
Mean ergodic theorems for weakly converging unitary representations and for strongly quasimixing dynamical systemsp. 127
Mean ergodic theorems for strongly quasimixing random fieldsp. 132
Mean ergodic theorem on SQM-spacesp. 132
Mean averaging nets on group extensions and productsp. 133
Construction of mean averaging sequences of sets on non-amenable groupsp. 135
Spectral criterionp. 139
Mean ergodic theorem for homogeneous generalized random fieldsp. 144
Mean ergodic theorem on the free group F[subscript 2]p. 149
Maximal and dominated ergodic theoremsp. 156
Preliminariesp. 156
Regular nets of setsp. 173
Hardy - Littlewood type inequalities for regular netsp. 183
Hardy - Littlewood type inequalities for some non-regular and singular weightsp. 185
Maximal and dominated ergodic theorems: The transfer approachp. 193
Maximal ergodic theorems: The dual space approachp. 195
Pointwise ergodic theoremsp. 207
Pointwise averaging nets: definition and preliminary discussionp. 207
Pointwise averaging in dense subsets of L[superscript 2]: Spectral approachp. 210
Pointwise averaging with regular ergodic nets of setsp. 217
Ergodic theorems with regular ergodic weightsp. 222
Pointwise averaging with spheres and spherical layersp. 224
Pointwise averaging with convolution weightsp. 226
General ratio ergodic theoremsp. 229
Construction of pointwise averaging sets on general connected groupsp. 234
Pointwise averaging nets on homogeneous spacesp. 241
Ergodic theorems for homogeneous random measuresp. 246
Homogeneous and [lambda]-homogeneous random fieldsp. 246
Local ergodic theorems. Lebesgue decomposition of homogeneous random measuresp. 249
"Global" ergodic theoremsp. 253
Homogeneous random measures on homogeneous spacesp. 259
Homogeneous random signed measuresp. 261
Specific informational and thermodynamical characteristics of homogeneous random fieldsp. 264
Main notations and definitionsp. 264
Generalized ergodic theoremsp. 268
Specific energyp. 271
Relative entropy and associated convergence theoremsp. 275
Generalizations of the McMillan theorem. Specific entropyp. 278
Specific free energyp. 286
Convergence theorems for Gibbsian measures. Specific relative entropyp. 287
The variational principlep. 291
Appendixp. 296
App. 1. Groups and semigroupsp. 296
App. 2. Homogeneous and group-type homogeneous spacesp. 309
App. 3. Amenable semigroups and ergodic netsp. 318
App. 4. Positive definite functionsp. 325
App. 5. Representations of semigroups in Banach spacesp. 328
App. 6. Weakly almost periodic elements and functionsp. 333
App. 7. Dynamical systemsp. 337
App. 8. Homogeneous random functionsp. 338
App. 9. Measurability and continuity of representations, dynamical systems and homogeneous random fieldsp. 343
App. 10. Flight-functions and FD-continuous spectrump. 349
App. 11. Relative entropy on random [sigma]-algebrasp. 353
App. 12. The Banach convergence principlep. 356
App. 13. Directions and netsp. 357
App. 14. Correspondence between "left" and "right" objects and conditionsp. 362
Referencesp. 364
Indexp. 394
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792317173
ISBN-10: 0792317173
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 399
Published: 1st July 1992
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 24.13 x 16.51  x 2.54
Weight (kg): 0.73