Dynamical notions along filters

TL;DR

This paper explores the localization of various dynamical concepts along filters, extending existing theories and establishing new properties, characterizations, and results on partition regularity of nonlinear equations in this context.

Contribution

It introduces generalized notions of syndetic and central sets along filters, provides nonstandard characterizations, and proves partition regularity of nonlinear equations under broad conditions.

Findings

Defined F-syndetic, F-central, and related sets along filters.

Established nonstandard characterizations of these notions.

Proved partition regularity of certain nonlinear equations along filters.

Abstract

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.