Xiaobo Liu
An earlier article with Francis Bonahon introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmuller space. We explicity compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.
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We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the intermediary steps are also done for general partially hyperbolic diffeomorphisms homotopic to the identity.
We announce some results towards the classification of partially hyperbolic diffeomorphisms on 3-manifolds, and outline the proofs in the case when the diffeomorphism is dynamically coherent. Detailed proofs are long and technical and will appear later.
We prove the existence of a minimal diffeomorphism isotopic to the identity between two hyperbolic cone surfaces $(Σ,g_1)$ and $(Σ,g_2)$ when the cone angles of $g_1$ and $g_2$ are different and smaller than $π$. When the cone angles of $g_1$ are strictly smaller than the ones of $g_2$, this minimal diffeomorphism is unique.
We establish a theory for the existence and regularity of solutions to the cohomological equation over an accessible, partially hyperbolic diffeomorphism. As a by-product of our techniques, we show that for $r>1$, any $C^r$ homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$ submanifold.
Partially hyperbolic diffeomorphisms that are center fixing
本文证明：若传递且动力学一致的偏双曲微分同胚具有一维中心叶状结构，并保持每个中心叶片不变，则其必为离散化 Anosov 流。
We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms $f$ on compact $3$-manifolds with fundamental groups of exponential growth such that $f^n$ is not homotopic to identity for all $n>0$. These provide counterexamples to a classification conjecture of Pujals.
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 work of Bonatti, Gogolev, Hammerlindl and Potrie to the whole isotopy class. We relate the techniques with the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in the previous paper by the authors. The appendix reviews some consequences of the Nielsen-Thurston classification of surface homeomorphisms to the dynamics of lifts of such maps to the universal cover.
A classification of partially hyperbolic diffeomorphisms on 3-dimensional manifolds with (virtually) solvable fundamental group is obtained. If such a diffeomorphism does not admit a periodic attracting or repelling two-dimensional torus, it is dynamically coherent and leaf conjugate to a known algebraic example. This classification includes manifolds which support Anosov flows, and it confirms conjectures by Rodriguez Hertz--Rodriguez Hertz--Ures and Pujals in the specific case of solvable fundamental group.
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 hyperbolic 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 ide…
Let $Σ_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(Σ_g)$ be the group of Hamiltonian diffeomorphisms of $Σ_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $Ham(Σ_g)$ is unbounded with respect to this metric.
Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Cauchy surface, hence extending Chernov-Nemirovski's observation to arbitrary dimensions.
Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part II: Branching foliations
研究同伦于恒等的3维部分双曲微分同胚，通过分析分支叶状结构，完成了Seifert纤维流形中的分类，并证明双曲流形中此类微分同胚或为动态相干且具离散化Anosov流幂，或属双平移类。
Partially Hyperbolic Dynamics with Quasi-isometric Center
本文研究了闭3-流形上具有拟等距中心的偏双曲微分同胚，证明了在无su-环面时的可达性与遍历性，确认了Hertz-Hertz-Ures遍历性猜想，并给出了此类微分同胚的完整分类。
Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is …
Partial hyperbolicity and pseudo-Anosov dynamics
本文证明容许部分双曲微分同胚的双曲 3 维流形必容许 Anosov 流，并利用中心分支叶理和坍缩 Anosov 流完整分类相关微分同胚。
We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples includ…
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