New York Journal of Mathematics
Volume 29 (2023), 792-817


Lorenzo Luperi Baglini, Sourav Kanti Patra, and Md Moid Shaikh

Dynamical notions along filters

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Published: June 30, 2023.
Keywords: Algebra in the Stone-Cech compactification, Dynamical system, F-uniform recurrence, F-proximality, JIFUR.
Subject [2010]: 37B20; 37B05; 05D10; 03H05.

We study the localization along a filter of several dynamical notions. This generalizes and extends similar localizations that have been considered in the literature, e.g. near 0 and near an idempotent. Definitions and basic properties of F-syndetic, piecewise F-syndetic, collectionwise F-piecewise syndetic, F-quasi central and F-central sets and their relations with F-uniformly recurrent points and ultrafilters are studied. We provide also the nonstandard characterizations of some of the above notions and we prove the partition regularity of several nonlinear equations along filters under mild general assumptions.


The authors are thankful to the anonymous referee for many suggestions that improved substantially the readability of the paper.

Author information

Lorenzo Luperi Baglini
Dipartimento di Matematica
Universitá di Milano, Via Saldini 50, 20133 Milan, Italy


Sourav Kanti Patra
Department of Mathematics
Centre for Distance And Online Education
The University of Burdwan
Purba Bardhaman, West Bengal, India


Md Moid Shaikh
Department of Mathematics
Maharaja Manindra Chandra College
20 Ramkanto Bose Street
Kolkata-700003, West Bengal, India