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1. Partially hyperbolic diffeomorphisms homotopicto the identity in dimension 3, II : Branching foliations

   https://doi.org/10.2140/gt.2023.27.3095  

   Barthelmé, Thomas; Fenley, Sérgio R; Frankel, Steven; Potrie, Rafael (January 2023, Geometry & Topology)

   We study 3–dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov’s center stable and center unstable branching foliations. This extends our previous study of the true foliations that appear in the dynamically coherent case. We complete the classification of such diffeomorphisms in Seifert fibered manifolds. In hyperbolic manifolds, we show that any such diffeomorphism is either dynamically coherent and has a power that is a discretized Anosov flow, or is of a new potential class called a double translation. 

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2. 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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3. Endperiodic automorphisms of surfaces and foliations

   https://doi.org/10.1017/etds.2019.56  

   CANTWELL, JOHN; CONLON, LAWRENCE; FENLEY, SERGIO R. (January 2021, Ergodic Theory and Dynamical Systems)

   We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations $${\mathcal{F}}$$ and $${\mathcal{F}}^{\prime }$$ are transverse to a common one-dimensional foliation $${\mathcal{L}}$$ whose monodromy on the non-compact leaves of $${\mathcal{F}}$$ exhibits the nice dynamics of Handel–Miller theory, then $${\mathcal{L}}$$ also induces monodromy on the non-compact leaves of $${\mathcal{F}}^{\prime }$$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends. 

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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. Polar foliations on symmetric spaces and mean curvature flow

   https://doi.org/10.1515/crelle-2022-0045  

   Liu, Xiaobo; Radeschi, Marco (August 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres. 

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