Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms

Abstract

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 Z 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.

Centralizers of partially hyperbolic diffeomorphisms in dimension 3

“…The proof in [DWX19] works equally well in any dimension. Here we point out that, if one considers only 3-manifolds, then some lemmas can be strengthened to obtain the rigidity result for a much broader class of partially hyperbolic diffeomorphisms.…”

“…In [DWX19], Damjanovic, Wilkinson and Xu investigate centralizers of certain partially hyperbolic diffeomorphisms and prove the following beautiful rigidity result: The centralizer of a perturbation of a time-1 map of an Anosov geodesic flow is either virtually Z or it is virtually R. In the latter case the partially hyperbolic diffeomorphism is the time-1 map of a smooth Anosov flow.…”

“…Remark 0.1. Note that Theorem B is a generalization of the 3-dimensional case of Theorem 3 of [DWX19]. (One has to take a power of f to obtain the embedding into an Anosov flow only in the case when M is a k-cover of the unit tangent bundle of a hyperbolic surface or an orbifold, see Remark 7.4 in [BFFP19]).…”

Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

“…The recent result of [11] is a good indication that the search of invariant metric system is an interesting tool or, sometimes, even the right tool. Still in the idea of future applications of our work it might be interesting to mention the possible use of our results in order to study the centralizers of a dynamics: the recent work of [15] illustrates once again the importante of having a dichotomy as we got.…”

Lyapunov stability and the Bernoulli property

“…The latter one shows that the center foliation is extremely pathological. Dichotomies of this type have been the key in several results, among which we cite [5] and a recent article of D. Damjanovic, A. Wilkinson and D. Xu [14] where this dichotomy is used to obtain a global classification of the centralizer of certain partially hyperbolic diffeomorphisms on three manifolds.…”