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Ergodic theory and connections with analysis and probability. (English) Zbl 0898.28005

There has been considerable progress in ergodic theory during the last 10-15 years due to new methods which utilize tools from harmonic analysis and probability. The principal purpose of this paper is to introduce the reader in a coherent way to these techniques. (Thus, many of the results have appeared elsewhere.) In sections 2 and 3 the author discusses tools which can help to show that for certain sequences of operators \(T_k\) and certain \(f\), \(T_kf\) diverges a.e. One of the most useful tools in this context is Bourgains entropy theorem. Section 4 discusses convolution powers of a single measure \(\mu\) on \(\mathbb{R}\) and Calderón’s transfer principle. Section 5 deals with good-lambda inequalities as introduced by Burkholder and Gundy. These kinds of inequalities have proved very useful in probability and analysis. Section 6 treats oscillation and variational inequalities. The proofs involve martingale theory. The variational inequalities give rize to inequalities for the number of \(\lambda\)-jumps that a sequence of ergodic averages can take. The results are too numerous to be stated here. Anybody working in this area or trying to enter it will find this article an illuminating introduction.

MSC:

28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
60G07 General theory of stochastic processes
40A05 Convergence and divergence of series and sequences
42A50 Conjugate functions, conjugate series, singular integrals
42B25 Maximal functions, Littlewood-Paley theory
60G42 Martingales with discrete parameter
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