  We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with
a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically
coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the
quotient space of the center foliation has the shadowing property, i.e. for
every $\varepsilon> 0$ there exists $\delta > 0$ such that every
$\delta$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of
center leaves. Although the shadowing orbit is not necessarily unique, we prove
the density of periodic center leaves inside the chain recurrent set of the
quotient dynamics. Some other interesting properties of the quotient dynamics
are discussed.
