Dynamics near an idempotent

Abstract

Hindman and Leader first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of ((0,∞),+). Using the algebraic structure of the Stone-Čech compactification, Tootkaboni and Vahed generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent e for a dense subsemigroup of a semitopological semigroup (R,+) and they gave the combinatorial proof of the Central Sets Theorem near e. Algebraically one can define quasi-central sets near e for dense subsemigroups of (R,+). In a dense subsemigroup of (R,+), C-sets near e are the sets, which satisfy the conclusions of the Central Sets Theorem near e. Patra gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near e. We also study minimal systems near e in the last section of this paper.