  Hindman and Leader first introduced the notion of semigroup of ultrafilters
converging to zero for a dense subsemigroups of $((0,\infty),+)$. Using the
algebraic structure of the Stone-$\breve{C}$ech compactification, Tootkabani
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 subsemigroups of a semitopological semigroup $(T, +)$ and they gave the
combinatorial proof of central set theorem near $e$. Algebraically one can also
define quasi-central sets near $e$ for dense subsemigroups of $(T, +)$. In a
dense subsemigroup of $(T,+)$, C-sets near $e$ are the sets, which satisfy the
conclusions of the central sets theorem near $e$. S. K. Patra gave dynamical
characterizations of these combinatorially rich sets near zero. In this paper
we shall prove these dynamical characterizations for these combinatorially rich
sets near $e$.
