Stable accessibility for partially hyperbolic diffeomorphisms is central to their ergodic theory, and we establish its \(C^1\)-density among 1. all, 2. volume-preserving, 3. symplectic, and 4. contact partially hyperbolic flows.
As applications, we obtain in each of these 4 categories \(C^1\)-density of \(C^1\)-stable topological transitivity, ergodicity, and triviality of the centralizer.
In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu (arXiv:1902.05201) who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in (arXiv:1908.06227).
We prove that for infinite rank-one transformations satisfying a property called "partial boundedness," the only commuting transformations are powers of the original transformation. This shows that a large class of infinite measure-preserving rank-one transformations with bounded cuts have trivial centralizers. We also characterize when partially bounded transformations are isomorphic to their inverse.
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 …
In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on $\mathbb{T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.
We show that for any integer $r \geq 2$, stable accessibility is $C^r$-dense among partially hyperbolic diffeomorphisms with two-dimensional center that satisfy some strong bunching and are stably dynamically coherent.
We consider classes of partially hyperbolic diffeomorphism $f:M\to M$ with splitting $TM=E^s\oplus E^c\oplus E^u$ and $\dim E^c=2$. These classes include for instance (perturbations of) the product of Anosov and conservative surface diffeomorphisms, skew products of surface diffeomorphisms over Anosov, partially hyperbolic symplectomorphisms on manifolds of dimension four with bidimensional center foliation whose center leaves are all compact. We prove that accessibility holds in these classes for $C^1$ open and $C^r$ dense subsets and moreover they are stably ergodic.
We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center dimension is one, we also give a criteria to obtain three-legged accessibility in a robust way. We show some applications of our results to the time-one map of Anosov flows, skew products and certain Anosov diffeomorphisms with partially hyperbolic splitting.
In this work we deal with partially hyperbolic diffeomorphisms whose central direction is two dimensional. We prove that in general the accessibility classes are immersed manifolds. If, furthermore, the diffeomorphism is dynamically coherent and satisfies certain bunching condition, then the accessibility classes are $C^{1}$ immersed manifolds.
This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and perturbations of Anosov flows under conditions on the dimensions of the invariant subbundles. The paper further shows that the non-open accessibility classes form a $C^1$ lamination and gives results about the accessibility classes of non-volume-preserving systems. Note: this document has been modified slightly from earlier preprints. The numbering of sections was chan…
We obtain a dichotomy for $C^1$-generic symplectomorphisms: either all the Lyapunov exponents of almost every point vanish, or the map is partially hyperbolic and ergodic with respect to volume. This completes a program first put forth by Ricardo Mañé. A main ingredient in our proof is a generalization to partially hyperbolic invariant sets of the main result in [Dolgopyat-Wilkinson] that stable accessibility is $C^1$ dense among partially hyperbolic diffeomorphisms.
We consider a class of partially hyperbolic diffeomorphisms introduced in [BFP] which is open and closed and contains all known examples. If in addition the diffeomorphism is non-wandering, then we show it is accessible unless it contains a su-torus. This implies that these systems are ergodic when they preserve volume, confirming a conjecture by Hertz-Hertz-Ures for this class of systems.
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Pierre Berger and Pablo Carrasco. This example is robustly non-uniformly hyperbolic, with two dimensional center, almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.
