567,538 research outputs found

    Variations on topological recurrence

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    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

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    The Khintchine recurrence theorem asserts that on a measure preserving system, for every set AA and ε>0\varepsilon>0, we have μ(A∩T−nA)≥μ(A)2−ε\mu(A\cap T^{-n}A)\geq \mu(A)^2-\varepsilon for infinitely many n∈Nn\in \mathbb{N}. We show that there are systems having under-recurrent sets AA, in the sense that the inequality μ(A∩T−nA)<μ(A)2\mu(A\cap T^{-n}A)< \mu(A)^2 holds for every n∈Nn\in \mathbb{N}. 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

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    Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a (r+2)(r+2)-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 rr 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

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    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

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    We study recurrence, and multiple recurrence, properties along the kk-th powers of a given set of integers. We show that the property of recurrence for some given values of kk 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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