Download PDF by Idris Assani: Ergodic Theory and Dynamical Systems

By Idris Assani

ISBN-10: 3110298139

ISBN-13: 9783110298130

This is often the complaints of the workshop on contemporary advancements in ergodic thought and dynamical platforms on March 2011 and March 2012 on the college of North Carolina at Chapel Hill. The articles during this quantity hide a number of features of shiny study in ergodic thought and dynamical structures. It includes contributions to Teichmuller dynamics, period alternate variations, persisted fractions, go back occasions averages, Furstenberg Fractals, fractal geometry of non-uniformly hyperbolic horseshoes, convergence alongside the series of squares, adic and horocycle flows, and topological flows. those contributions illustrate the connections among ergodic concept and dynamical structures, quantity conception, harmonic research, chance, and algebra. surveys are incorporated which offer a pleasant creation for younger or senior researcher to a couple energetic learn parts. total this quantity offers a truly precious mix of innovations and techniques in addition to instructions of study on normal convergence phenomena in ergodic thought and dynamical platforms.

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1 , ????2 ), lim ???? 1 ????−1 ∑ ????(???????? (????1 ))????(???????? (????2 )) = 0. ???? ????=0 The link between product-null functions, spectral measure and the Kronecker fac­ tor is outlined in the following lemma of Rudolph. Note that this lemma actually cor­ rects the corresponding statement in the BFKO proof. 22. For ???? ∈ ????∞ (????) with ∫ ???????????? = 0, the following are equivalent: 34 | Idris Assani and Kimberly Presser (1) (2) (3) (4) ???? is product-null. The spectral measure of ???? is nonatomic. e. ???? ∈ K⊥ . e. ergodic component ????̂ of ???? × ???? ∫ ????(????1 )????(????2 )????????̂ = 0.

Unfortunately, there was an error in the proof of Wiener and Wintner. However, the theorem is true and has been proven multiple times using a wide variety of techniques. The first correct version of the proof was given by H. Furstenberg [50] using the notions of joinings and generic points. We will see later the role of joinings and generic points in the proof of the Return Times Theorem. Another proof of the Wiener–Wintner Theorem which uses Van der Corput’s inequality [60] was given by Bourgain [27] in 1990.

1 , ????2 ), lim ???? 1 ????−1 ∑ ????(???????? (????1 ))????(???????? (????2 )) = 0. ???? ????=0 The link between product-null functions, spectral measure and the Kronecker fac­ tor is outlined in the following lemma of Rudolph. Note that this lemma actually cor­ rects the corresponding statement in the BFKO proof. 22. For ???? ∈ ????∞ (????) with ∫ ???????????? = 0, the following are equivalent: 34 | Idris Assani and Kimberly Presser (1) (2) (3) (4) ???? is product-null. The spectral measure of ???? is nonatomic. e. ???? ∈ K⊥ . e. ergodic component ????̂ of ???? × ???? ∫ ????(????1 )????(????2 )????????̂ = 0.

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