In this paper, we discuss the notions of $G$-attractor and $G$-expansive. It is found that if $<(X,f_n)>$ is a sequence of dynamical systems converging $G$-uniformly to $f$ and if the sequence $<(X,f_n)>$ has a $G$-uniform attractor $Y\subset X$, then $Y$ is also a $G$-attractor of $f$. We also show that if $<(X,f_n)>$ is a sequence of $G$-expansive dynamical systems with same expansivity time and expansivity constant and converging $G$-uniformly to $f$, then $(X,f)$ is also $G$-expansive. We investigate the $G$-mixing, $G$-sensitive and $G$-shadowing property of the orbital limit $f$.
Keywords and Phrases
$G$-Attractor, $G$-Expansive, $G$-Sensitive, $G$-Mixing, $G$-Shadowing property, $G$-Nonwandering.
A.M.S. subject classiﬁcation
37B05, 37C85, 54A20.