Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics

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 $F:M\mathrm{/}{F}^c\to M\mathrm{/}{F}^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\epsilon >0$ there exists $\delta >0$ such that every $\delta$-pseudo orbit of center leaves is $\epsilon$-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.