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1. Partially hyperbolic diffeomorphisms homotopic to to the identity in dimension 3 Part I: The dynamically coherent case

   Barthelme, Thomas; Fenley, Sergio; Frankel, Steven; Potrie, Rafael (January 2024, Annales Scientifiques de lEcole Normale Supérieure)

   We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center-stable and center-unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyper- bolic or Seifert-fibered 3-manifold is leaf-conjugate to the time-one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz– Hertz–Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds. 

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2. Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

   https://doi.org/10.1017/etds.2020.113  

   BARTHELMÉ, THOMAS; FENLEY, SERGIO R.; FRANKEL, STEVEN; POTRIE, RAFAEL (November 2021, Ergodic Theory and Dynamical Systems)

   Abstract 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. 

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3. Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

   https://doi.org/10.1017/etds.2021.42  

   AVILA, A.; VIANA, MARCELO; WILKINSON, A. (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact. 

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4. Cartan actions of higher rank abelian groups and their classification

   https://doi.org/10.1090/jams/1033  

   Spatzier, Ralf; Vinhage, Kurt (July 2024, Journal of the American Mathematical Society)

   We study ${\mathbb{R}}^k\times <\#comment/>{\mathbb{Z}}^{\mathrm{\ell <\#comment/>}}$actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. 

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5. Partial hyperbolicity and pseudo-Anosov dynamics

   https://doi.org/10.1007/s00039-024-00670-1  

   Fenley, Sergio; Potrie, Rafael (January 2024, Geometric and functional analysis)

   We study partial hyperbolic (PH) diffeomorphisms in 3-manifolds satisfying a commuting property. This is mainly applicable when the manifold is either hyperbolic or Seifert. As a consequence we prove that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism, then it also admits an Anosov flow. 

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