591,905 research outputs found
Variations on topological recurrence
Recurrence properties of systems and associated sets of integers that suffice
for recurrence are classical objects in topological dynamics. We describe
relations between recurrence in different sorts of systems, study ways to
formulate finite versions of recurrence, and describe connections to
combinatorial problems. In particular, we show that sets of Bohr recurrence
(meaning sets of recurrence for rotations) suffice for recurrence in
nilsystems. Additionally, we prove an extension of this property for multiple
recurrence in affine systems
Under recurrence in the Khintchine recurrence theorem
The Khintchine recurrence theorem asserts that on a measure preserving
system, for every set and , we have for infinitely many . We show that there
are systems having under-recurrent sets , in the sense that the inequality
holds for every . In
particular, all ergodic systems of positive entropy have under-recurrent sets.
On the other hand, answering a question of V.~Bergelson, we show that not all
mixing systems have under-recurrent sets. We also study variants of these
problems where the previous strict inequality is reversed, and deduce that
under-recurrence is a much more rare phenomenon than over-recurrence. Finally,
we study related problems pertaining to multiple recurrence and derive some
interesting combinatorial consequences.Comment: 18 pages. Referee's comments incorporated. To appear in the Israel
Journal of Mathematic
Computing recurrence coefficients of multiple orthogonal polynomials
Multiple orthogonal polynomials satisfy a number of recurrence relations, in
particular there is a -term recurrence relation connecting the type II
multiple orthogonal polynomials near the diagonal (the so-called step-line
recurrence relation) and there is a system of recurrence relations
connecting the nearest neighbors (the so-called nearest neighbor recurrence
relations). In this paper we deal with two problems. First we show how one can
obtain the nearest neighbor recurrence coefficients (and in particular the
recurrence coefficients of the orthogonal polynomials for each of the defining
measures) from the step-line recurrence coefficients. Secondly we show how one
can compute the step-line recurrence coefficients from the recurrence
coefficients of the orthogonal polynomials of each of the measures defining the
multiple orthogonality.Comment: 22 pages, 2 figures in Numerical Algorithms (2015
How to avoid potential pitfalls in recurrence plot based data analysis
Recurrence plots and recurrence quantification analysis have become popular
in the last two decades. Recurrence based methods have on the one hand a deep
foundation in the theory of dynamical systems and are on the other hand
powerful tools for the investigation of a variety of problems. The increasing
interest encompasses the growing risk of misuse and uncritical application of
these methods. Therefore, we point out potential problems and pitfalls related
to different aspects of the application of recurrence plots and recurrence
quantification analysis
Powers of sequences and recurrence
We study recurrence, and multiple recurrence, properties along the -th
powers of a given set of integers. We show that the property of recurrence for
some given values of does not give any constraint on the recurrence for the
other powers. This is motivated by similar results in number theory concerning
additive basis of natural numbers. Moreover, motivated by a result of Kamae and
Mend\`es-France, that links single recurrence with uniform distribution
properties of sequences, we look for an analogous result dealing with higher
order recurrence and make a related conjecture.Comment: 30 pages. Numerous small changes made. To appear in the Proceedings
of the London Mathematical Societ
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