Ergodic TheoryThe study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated. |
Contents
II | 8 |
IV | 8 |
VI | 8 |
VII | 8 |
IX | 8 |
XIII | 9 |
XV | 10 |
XVIII | 11 |
LIX | 153 |
LX | 156 |
LXI | 158 |
LXII | 162 |
LXIII | 167 |
LXIV | 175 |
LXV | 181 |
LXVI | 186 |
XXI | 12 |
XXIII | 13 |
XXV | 14 |
XXVI | 15 |
XXVII | 17 |
XXVIII | 19 |
XXIX | 20 |
XXX | 23 |
XXXII | 27 |
XXXIII | 33 |
XXXIV | 41 |
XXXV | 57 |
XXXVI | 64 |
XXXVII | 74 |
XXXIX | 75 |
XL | 76 |
XLI | 87 |
XLII | 90 |
XLIV | 93 |
XLV | 100 |
XLVI | 103 |
XLVII | 107 |
XLVIII | 113 |
XLIX | 119 |
L | 126 |
LI | 133 |
LIV | 135 |
LV | 141 |
LVI | 150 |
LVIII | 151 |
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Common terms and phrases
A₁ abelian group automorphisms B₁ B₂ Bernoulli schemes Bernoulli shift Borel cascade choose constant a.e. Convergence Theorem converges a.e. Corollary countable cylinder sets defined denote dense dense set disjoint eigenfunction eigenvalue entropy equivalence classes ergodic m.p.t. example Exercise filler finite measure finite partition fixed Furstenberg given Haar measure hence homeomorphism implies infinitely integral invariant isometry isomorphic Kakutani L²(X Lebesgue space Lemma lim sup log₂ Markov shift Maximal Ergodic Theorem maximal inequality measurable partition measurable sets measure algebra measure space metric space minimal o-algebra orbit Ornstein P₁ periodic sequence pointwise probability measure probability space Proof Proposition prove sets of measure skeleton strongly mixing sub-o-algebras subset Suppose T-invariant T₁ theory topological topological entropy transformation uniquely ergodic weak mixing weakly mixing X₁