維納一溫特納遍歷定理WIENER WINTNER ERGODIC THEOREMS

內(nèi)容概要

The Wiener Wintner ergodic theorem is a strengthening of Birkhoff pointwise ergodic theorem. Announced by N Wiener and A Wintner, this theorem has introduced the study of a general phenomenon in ergodic theory in which samplings are "good" for an uncountable number of systems. We study the rate of convergence in the uniform version of this theorem and what we call Wiener Wintner dynamical systems and prove for these systems two pointwise results: the a.e. double recurrence theorem and the a.e. continuity of the fractional rotated ergodic Hilbert transform. Some extensions of the Wiener Wintner ergodic theorem are also given.

書(shū)籍目錄

Chapter 1  The Mean and Pointwise Ergodic Theorems  1.1  The mean ergodic theorem  1.2  The pointwise ergodic theorem    1.2.1  Birkhoff's ergodic theorem through the maximal inequality    1.2.2  Birkhoff's ergodic theorem with no maximal inequality    1.2.3  Maximal inequalities, dominated ergodic theorem and transference    1.2.4  The pointwise ergodic theorem through a variational inequalityChapter 2  Wiener Wintner Pointwise Ergodic Theorems  2.1  Introduction  2.2  Ergodic transformations, Kronecker factors, spectral measures  2.3  Wiener Wintner theorem through the affinity of measures     2.3.1  Preliminaries on sequences having a correlation and the affinity    2.3.2  First proof of the Wiener Wintner ergodic theorem   2.4  Wiener Wintner theorem through a simple inequality    2.4.1  A simple variant of Van der Corput's inequality    2.4.2  J. Bourgain's uniform Wiener Wintner ergodic theorem  2.5  Wiener Wintner theorem through disjointness    2.5.1  Disjointness and generic points    2.5.2  The third proof  2.6  Topological Wiener Wintner ergodic theorem    2.6.1  Topological dynamical systems    2.6.2  Wiener Wintner results for uniquely ergodic systems   2.7  Remarks and questions    2.7.1  Ergodic decomposition    2.7.2  Comments    2.7.3  RemarksChapter 3  Universal Weights for Dynamical Systems  3.1  Introduction  3.2  Independent variables as universal weights for the pointwise ergodic theorem    3.2.1  Independent variables as universal weights for the pointwise convergence in L2    3.2.2  Independent random variables as universal weights for the pointwise convergence in LpChapter 4  J. Bourgain~s Return Times Theorem  4.1  Introduction  4.2  Preliminaries  4.3  A proof of the return times theorem    4.3.1  Proof for f x    4.3.2  Proof for f x      4.3.2.1   Continuity of the spectral measure of f       4.3.2.2  The finite range assumption      4.3.2.3  Part I      4.3.2.4  Part II      4.3.2.5  Part III      4.3.2.6  Part IV      4.3.2.7  Part V      4.3.2.8  The contradiction      4.3.2.9  Extending beyond the finite range assumption  ……Chapter 5 Extensions of the Return Times TheoremChapter 6 Speed of Convergence in the Uniform Wiener Wintner TheoremChapter 7 Weak Wiener Wintner Dynamical SystemsChapter 8 Polynomial Wiener Wintner Ergodic TheoremChapter 9 Extension to More General OperatorsBibliographyIndex

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