Denseness of ergodicity for a class of partially hyperbolic volume-preserving flows

无周期点的部分双曲微分同胚的可及性与遍历性

Accessibility and Ergodicity of Partially Hyperbolic Diffeomorphisms without Periodic Points

本文证明了无周期点的闭3-流形上$C^2$保守部分双曲微分同胚的遍历性，并给出了无周期点情形下Hertz-Hertz-Ures遍历性猜想的肯定回答，同时在非游荡集为全空间且基本群非虚拟可解时证明了其可及性。

Partial hyperbolicity and ergodicity in dimension three

In [15] the authors proved the Pugh-Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e. stable ergodic diffeomorphism are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.

Stable Ergodicity and Accessibility for certain Partially Hyperbolic Diffeomorphisms with Bidimensional Center Leaves

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.

The local $C^1$-density of stable ergodicity

The center bundle of a conservative partially hyperbolic diffeomorphism $f$ is called robustly non-hyperbolic if any conservative diffeomorphism which is $C^1$-close to $f$ has non-hyperbolic center bundle. In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems with robust non-hyperbolic center.

关于双曲自同态的旋转扩张的遍历性

On the Ergodicity of Rotation Extensions of Hyperbolic Endomorphisms

研究部分双曲自同态的遍历性，通过可达性证明斜积的遍历性与拓扑传递性，并给出中心维数1时的遍历稳定性结果。

$C^{r}-$prevalence of stable ergodicity for a class of partially hyperbolic systems

We prove that for $r \in \mathbb{N}_{\geq 2} \cup \{\infty\}$, for any dynamically coherent, center bunched and strongly pinched volume preserving $C^r$ partially hyperbolic diffeomorphism $f \colon X \to X$, if either (1) its center foliation is uniformly compact, or (2) its center-stable and center-unstable foliations are of class $C^1$, then there exists a $C^1$-open neighbourhood of $f$ in ${\rm Diff}^r(X,\mathrm{Vol})$, in which stable ergodicity is $C^r$-prevalent in Kolmogorov's sense. In particular, we verify Pugh-Shub's stable ergodicity conjecture in this region. This also provides t…

Ergodicity and partial hyperbolicity on Seifert manifolds

We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms which do not have any hyperbolic invariant subbundle. Moreover the uniqueness of SRB measures for non-conservative $C^1$ perturbations of such diffeomorphisms. This class contains strictly non-partially hyperbolic robustly transitive diffeomorphisms by Bonatti-Viana and so we answer their question about the stable ergodicity of such systems.

Ergodicity and partial hyperbolicity on the 3-torus

If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism $A$, it is topologically conjugate to $A$.

Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

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.

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 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.

零熵非周期ergodic测度的稠密性

Denseness of zero entropy aperiodic ergodic measures

研究部分双曲同宿类中零维中心包的C^1-典型微分同胚的稠密性，证明零熵低复杂度测度在相应测度集中的稠密性。

Random permutations and unique fully supported ergodicity for the Euler adic transformation

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

On the ergodic properties of time changes of partially hyperbolic homogeneous flows

For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.

On the abundance of non-zero central Lyapunov exponents, physical measures and stable ergodicity for partially hyperbolic dynamics

We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.

Conservative partially hyperbolic dynamics

We discuss recent progress in understanding the dynamical properties of partially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are density of stable ergodicity and stable accessibility, center Lyapunov exponents, pathological foliations, rigidity, and the surprising interrelationships between these notions.

三维部分双曲微分同胚、遍历性与横截叶状结构

Partially hyperbolic diffeormorphisms, ergodicity, and transverse foliations in dimension 3

本文完成Pujals计划，基于Anosov流给出三维流形中（链）传递部分双曲微分同胚的完整拓扑分类，解答Hertz-Hertz-Ures的遍历性猜想，其定理B将在后续文章中推广以移除链传递假设。

A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms

In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the $C^1$ topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.

A survey on partially hyperbolic dynamics

Some of the guiding problems in partially hyperbolic systems are the following: (1) Examples, (2) Properties of invariant foliations, (3) Accessibility, (4) Ergodicity, (5) Lyapunov exponents, (6) Integrability of central foliations, (7) Transitivity and (8) Classification. Here we will survey the state of the art on these subjects, and propose related problems.

$C^1$ density of stable ergodicity

We prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for the $C^1$ topology: linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.

三维中同伦于恒等的部分双曲微分同胚

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three

本文证明了三维中同伦于恒等的保守部分双曲微分同胚在环境流形基本群非虚可解时具有可达性，从而证实了该同伦类下的Hertz-Hertz-Ures遍历性猜想。

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.

Rigidity for partially hyperbolic diffeomorphisms

In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute…

三维空间中的传递部分双曲微分同胚

Transitive partially hyperbolic diffeomorphisms in dimension three

该研究证明了三维闭合流形上C^{1+α}保守部分双曲微分同胚的传递性与遍历性等价，并给出了可访问类的精确分类。

Accessibility and ergodicity for collapsed Anosov flows

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.

部分双曲流的可达性与中心化子

Accessibility and centralizers for partially hyperbolic flows

本文证明了部分双曲流中稳定可达性的$C^1$稠密性，并由此推导出四类流中拓扑传递性、遍历性及中心化子平凡性的$C^1$稠密性。

具有拟等距中心的偏双曲动力学

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。

通过共形形变构造部分双曲测地流

Partially hyperbolic geodesic flow via conformal deformation

本文利用共形形变构造非 Anosov 部分双曲测地流，明确所需条件，并得到非正曲率、遍历性、最大熵测度唯一性及含共轭点的例子。