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Mixing property and pseudo random sequences. (English) Zbl 1132.11042

Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 189-197 (2006).
The author considers piecewise monotone \(C^{2}\)-transformations \(F\) on the unit interval \(I= [0,1].\) To such transformations a Perron-Frobenius operator is associated. It is well known the spectra of this operator determine the ergodic properties of the dynamical system \((I, F)\). Using such transformations generalized van der Corput sequences are defined and those sequences are determined which satisfy a discrepancy bound \(D_{N}= O(\frac{\log N}{N}).\) Furthermore, extensions to higher dimensions are discussed.
For the entire collection see [Zbl 1113.60008].

MSC:

11K45 Pseudo-random numbers; Monte Carlo methods
37B10 Symbolic dynamics
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