  We discover a rigidity phenomenon within the volume-preserving partially
hyperbolic diffeomorphisms with $1$-dimensional center. In particular, for
smooth, ergodic perturbations of certain algebraic systems -- including the
discretized geodesic flows over hyperbolic manifolds and certain toral
automorphisms with simple spectrum and exactly one eigenvalue on the unit
circle, the smooth centralizer is either virtually $\mathbb Z^\ell$ or contains
a smooth flow.
  At the heart of this work are two very different rigidity phenomena. The
first was discovered in [2,3] for a class of volume-preserving partially
hyperbolic systems including those studied here, the disintegration of volume
along the center foliation is either equivalent to Lebesgue or atomic. The
second phenomenon is the rigidity associated to several commuting partially
hyperbolic diffeomorphisms with very different hyperbolic behavior transverse
to a common center foliation [25].
  We introduce a variety of techniques in the study of higher rank, abelian
partially hyperbolic actions: most importantly, we demonstrate a novel
geometric approach to building new partially hyperbolic elements in hyperbolic
Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat
measure rigidity for circle extensions of Anosov diffeomorphisms and apply
normal form theory to upgrade regularity of the centralizer.

Dynamical Systems