08 Şubat 2016,
Sabancı Üniversitesi Matematik Bölümü Seminerleri
Unique ergodicity and distribution functions for subsequences of the van der Corput sequence
Polish Academy of Sciences, Polonya
Ergodic theory concerns with the behavior of a dynamical system when it is allowed to run for a long time. The theory is a modern approach in number theory which can be
used to study and characterize sets of numbers from a probabilistic or measure-theoretic point of view. As it is the main tool in the talk, I will first give a quick introduction to the main idea of the subject. In the main talk, I would like to present a joint work, with R. Nair, on the characterization of unique ergodicity based on certain subsequences of the natural numbers, called Hartman uniformly distributed sequences. Then we shall see an application of this characterization theorem on answering a question of O. Strauch, but in a more general framework, regarding the limit distribution of consecutive elements of the van der Corput sequence. Recently, C. Aistleitner and M. Hofer calculated the asymptotic distribution function of (Φb(n),Φb(n + 1),....,Φb(n + s -1))) n=0 to 1 on [0; 1)^s; where (Φb(n))n=0 to 1 denotes the van der Corput sequence in base b > 1; and showed that it is a copula. In the talk, we shall see that this phenomenon extends not only to a broad class of subsequences of the van der Corput sequence but also to a more general setting in the a-adic numbers. Indeed, we shall use the characterization of unique ergodicity, together with the fact that the van der Corput sequence can be seen as the orbit of the origin under the ergodic Kakutani-von Neumann transformation. Incidentally, we have introduced the a-adic van der Corput sequence which significantly generalizes the classical van der Corput sequence. I will go a bit further to present a new joint work, with A. Jaššová and R. Nair, on extending the a-adic van der Corput sequence to the Halton sequence in a generalized numeration system. We shall see that it provides a wealth of low-discrepancy sequences, which are very important in the quasi-Monte Carlo method. Note that some knowledge of measure theory and some interest in uniform distribution theory of sequences should be enough to follow the talk.
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