  Let $(X,G,\Phi)$ be a dynamical system, where $X$ is compact Hausdorff space,
and $G$ is a countable discrete group. We investigate shadowing property and
mixing between subshifts and general dynamical systems. For the shadowing
property, fix some finite subset $S\subset G$. We prove that if $X$ is totally
disconnected, then $\Phi$ has $S$-shadowing property if and only if
$(X,G,\Phi)$ is conjugate to an inverse limit of a sequence of shifts of finite
type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric
space (may be not totally disconnected), we prove that if $\Phi$ has
$S$-shadowing property, then $(X,G,\Phi)$ is a factor of an inverse limit of a
sequence of shifts of finite type by a factor map which almost lifts
pseudo-orbit for $S$.
  On the other hand, let property $P$ be one of the following property:
transitivity, minimal, totally transitivity, weakly mixing, mixing, and
specification property. We prove that if $X$ is totally disconnected, then
$\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is conjugate to an inverse
limit of an inverse system that consists of subshifts with property $P$ which
satisfies Mittag-Leffler condition. Also, for the case of metric space (may be
not totally disconnected), if property $P$ is not minimal or specification
property, we prove that $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is
a factor of an inverse limit of a sequence of subshifts with property $P$ which
satisfies Mittag-Leffler condition.
