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JMD

We consider a partially hyperbolic $C^1$-diffeomorphism $f\colon M
\rightarrow M$ with a uniformly compact $f$-invariant center foliation
$\mathcal{F}^c$. We show that if the unstable bundle is
one-dimensional and oriented, then the holonomy of the center
foliation vanishes everywhere, the quotient space $M/\mathcal{F}^c$ of
the center foliation is a torus and $f$ induces a hyperbolic
automorphism on it, in particular, $f$ is centrally transitive.

We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).

We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy-invariant family of measures on the unstable leaves (Margulis measure).

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