Centralizers of derived-from-Anosov systems on : rigidity versus triviality

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.

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

<p style='text-indent:20px;'>We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Möbius band or a plane.</p><p style='text-indent:20px;'>Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-map (for <inline-formula><tex-math id="M2">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula> large) of a non-transitive Anosov flow <inline-formula><tex-math id="M3">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula> on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to <inline-formula><tex-math id="M4">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula>. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.</p>

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms

Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphism f. We define the unstable pressure $P^{u}(f, \varphi )$ of f at a continuous function $\varphi $ via the dynamics of f on local unstable leaves. A variational principle for unstable pressure $P^{u}(f, \varphi )$ , which states that $P^{u}(f, \varphi )$ is the supremum of the sum of the unstable entropy and the integral of $\varphi $ taken over all invariant measures, is obtained. U-equilibrium states at which the supremum in the variational principle attains and their relation to Gibbs u-states are studied. Differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Fréchet differentiability and their relations to u-equilibrium states, are also considered.

Classification of partially hyperbolic diffeomorphisms under some rigid conditions

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

Codimension one compact center foliations are uniformly compact

Let $f:M\rightarrow M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.

Local unstable entropies of partially hyperbolic diffeomorphisms

Consider a $C^{1}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$. Following the ideas in establishing the local variational principle for topological dynamical systems, we introduce the notions of local unstable metric entropies (and local unstable topological entropy) relative to a Borel cover ${\mathcal{U}}$ of $M$. It is shown that they coincide with the unstable metric entropy (and unstable topological entropy, respectively), when ${\mathcal{U}}$ is an open cover with small diameter. We also define the unstable tail entropy in the sense of Bowen and the unstable topological conditional entropy in the sense of Misiurewicz, and demonstrate that both of them vanish. Some generalizations of these results to the case of unstable pressure are also investigated.

On the stable ergodicity of Berger–Carrasco’s example

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a 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.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.