Christian Bonatti, Sylvain Crovisier
We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects K at exactly one point.
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In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the $u$-equilibrium states are given.
We address the problem of existence and uniqueness (finite- ness) of ergodic equilibrium states for a natural class of partially hyperbolic dynamics homotopic to Anosov. We propose to study the disintegration of equilibrium states along central foliation as a tool to develop the theory of equilibrium states for partially hyperbolic dynamics.
Let $\mathcal{F}$ be a $C^2$ random partially hyperbolic dynamical system. For the unstable foliation, the corresponding unstable metric entropy, unstable topological entropy and unstable pressure via the dynamics of $\mathcal{F}$ on the unstable foliation are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem for unstable metric entropy is given, and a variational principle for unstable pressure (and hence for unstable entropy) is obtained. Moreover, as an application of the variational principle, equilibrium states for the unstable pressure including Gibbs $u$-states …
Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms
本文提出了“纤维化提升部分双曲微分同胚”的概念，证明了其同痕类内的部分双曲微分同胚具有动力学连贯性，并得出了同一连通分支内系统叶共轭的全局稳定性结果。
Katok's entropy formula is an important formula in entropy theory. This paper is devoted to establishing Katok's entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.
We use the Invariance Principle of Avila and Viana to prove that every partially hyperbolic symplectic diffeomorphism with 2-dimensional center bundle, and satisfying certain pinching and bunching conditions, can be $C^r$-approximated by non-uniformly hyperbolic diffeomorphisms.
We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms.
Pugh and Shub have conjectured that essential accessibility implies ergodicity, for a $C^2$, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satsified by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for $C^{1+γ}$ partially hyperbolic systems.
We prove the saturation of a generalized partially hyperbolic attractor of a $C^2$ map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a $C^1$-generic diffeomorphism has zero volume. In contrast, by modification of Poincaré cross section of the geometric model, we build a $C^1$-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume.
Ergodicity Of Partially Hyperbolic Endomorphisms
本文证明了在保体积、部分双曲、中心丛状且具有常Jacobian的条件下，自同态的本质可达性可推出其遍历性。
In this paper, we introduce the unstable topological pressure for C^1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. Moreover, without any additional assumption, we have established the expected variational principle which connects this unstable topological pressure and the unstable measure theoretic entropy, as well as the corresponding Lyapunov exponent.
We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume.
Expansive partially hyperbolic diffeomorphisms with one-dimensional center
本研究针对具有一维中心的扩张部分双曲微分同胚，给出了其成为拓扑Anosov的充分条件，推进了动力系统领域相关理论研究。
We show that there are no partially hyperbolic horseshoes with positive Lebesgue measure for diffeomorphisms whose class of differentiability is higher than 1. This generalizes a classical result by Bowen for uniformly hyperbolic horseshoes.
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.
We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable …
We show that in the absence of periodic centre annuli, a partially hyperbolic surface endomorphism is dynamically coherent and leaf conjugate to its linearisation. We proceed to characterise the dynamics in the presence of periodic centre annuli. This completes a classification of partially hyperbolic surface endomorphisms.
We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.
In this paper, we construct a partially hyperbolic skew-product diffeomorphism $f$ on $\mathbb{T}^3$, such that $f$ is accessible and chain transitive, but not transitive.
For every $r\in\mathbb{N}_{\geq2}\cup\{\infty\}$, we prove the $C^r$-closing lemma for general and conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle. In particular, it implies periodic points are dense for $C^r$-generic conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle.
We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^2$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^2$ which does not admit a centre foliation.
Physical measures for partially hyperbolic diffeomorphisms with mixed hyperbolicity
本文针对中心方向满足 u-定性质的混合双曲部分双曲微分同胚，证明其仅有有限多个物理测度，且这些测度的吸引盆在环境空间中覆盖满 Lebesgue 测度子集。
We prove results related to robust transitivity and density of periodic points of Partially Hyperbolic Diffeomorphisms under conditions involving Accessibility and a property in the tangent bundle .
We study a class of two dimensional partially hyperbolic systems, not necessarily skew products, trying to establish the germ of a general theory. To illustrate the scope of the theory, we apply our results to the case of fast-slow partially hyperbolic systems.
We prove that if $f$ is a partially hyperbolic diffeomorphism on 3-torus $\mathbb{T}^3$ and isotopic to Anosov, then $f$ is chain transitive.
We prove that at least one of the two invariant laminations of a strongly partially hyperbolic attractor with one-dimensional center bundle is minimal. This result extends those in [7] about minimal foliations for robustly transitive diffeomorphisms.
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in \cite{Bo, PS2007}. Our results give new insights to the multi…
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.
In these notes we prove that the $s$ or $u$-states of cocycles over partially hyperbolic maps are closed in the space of invariant measures.
Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms
本文研究了连续wise超几何同胚的稳定/不稳定同宿线的正则性，证明了在特定条件下周期点的密度以及同胚的transitivity。
We discuss about the denseness of the strong stable and unstable manifolds of partially hyperbolic diffeomorphisms. In this sense, we introduce a concept of m-minimality. More precisely, we say that a partially hyperbolic diffeomorphisms is m-minimal if m-almost every point in M has its strong stable and unstable manifolds dense in M. We show that this property has dynamics consequences: topological and ergodic. Also, we prove the abundance of m-minimal partially hyperbolic diffeomorphisms in the volume preserving and symplectic scenario.
Partially hyperbolic symplectomorphism with C^1 bundles
本文证明了四维部分双曲辛同胚在稳定丛和不稳定丛为C^1时的动力学相干性。
We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new technique with other constructions, we prove that $C^1$-generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.
Homoclinic points of symplectic partially hyperbolic systems with 2D centre
本文研究一类通有辛部分双曲微分同胚，证明其中每个双曲周期点都具有横截同宿点。
Unique equilibrium states for some partially hyperbolic diffeomorphisms with dominated splittings
本文研究带支配分裂的部分双曲微分同胚，在 Hölder 连续势函数振荡较小时证明平衡态的唯一性与鲁棒性。
We show that a strong partially hyperbolic diffeomorphism of $\mathbb{T}^3$ isotopic to Anosov has a unique quasi-attractor. Moreover, we study the entropy of the diffeomorphism restricted to this quasi-attractor.
We show that for any integer $r \geq 2$, stable accessibility is $C^r$-dense among partially hyperbolic diffeomorphisms with two-dimensional center that satisfy some strong bunching and are stably dynamically coherent.
We show that every partially hyperbolic diffeomorphism with a 1-dimensional center bundle has a principal symbolic extension. On the other hand, we show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing non-hyperbolic robustly transitive sets with center indecomposable bundle of dimension at least 2.
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.
An upper bound for the box dimension of the hyperbolic dynamics via unstable pressure
该研究利用次可加不稳定压强推导C¹双曲集上盒维数的上界，同时给出拓扑压强的新表达式，丰富了双曲动力学相关理论。
The purpose of this article is to obtain dynamically coherence of partially hyperbolic diffeomorphisms in certain classes of Anosov diffeomorphisms on nilmanifolds, extending a result due to T. Fisher, R. Potrie and M. Sambarino [FPS] on the torus.
We prove that stable ergodicity is C r open and dense among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, for all r in [2,infty]. The proof follows Pugh-Shub program: among conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle, accessibility is C r open and dense, and essential accessibility implies ergodicity.
Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a compact Riemannian manifold $M$, in this paper we define the concept of unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily compact and we extend a theorem of R. Bowen proving that, for an ergodic $f$-invariant measure $μ$, the unstable measure theoretical entropy of $f$ is upper bounded by the unstable topological entropy of $f$ on any set of full $μ$-measure. We also define a notion of unstable topological entropy of $f$ using a Hausdorff dimension like characterization.
We show that for a large space of volume preserving partially hyperbolic diffeomorphisms of the 3-torus with non-compact central leaves the central foliation generically is non-absolutely continuous.
We prove the existence and uniqueness of the SRB measure for singular partially hyperbolic flow in dimension $d\ge 3$. The proof does not use Poincaré sectional maps, but uses basic properties of thermodynamic formalism.
Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems
本文利用动力学密度基研究部分双曲集拓扑结构，证明正体积强部分双曲集包含全局稳定与不稳定流形，并给出关于本质可达、可达且中心丛聚条件下ACIP存在性与传递性的若干应用。
It was shown that in robustly transitive, partially hyperbolic diffeomorphisms on three dimensional closed manifolds, the strong stable or unstable foliation is minimal. In this article, we prove ``almost all'' leaves of both stable and unstable foliations are dense in the whole manifold.
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。
We survey a collection of recent results on center Lyapunov exponents of partially hyperbolic diffeomorphisms. We explain several ideas in simplified setups and formulate the general versions of results. We also pose some open questions.
In this work we study relations between regularity of invariant foliations and Lyapunov exponents of partially hyperbolic diffeomorphisms. We suggest a new regularity condition for foliations in terms of desintegration of Lebesgue measure which can be considered as a criterium for rigidity of Lyapunov exponents.
Simple Lyapunov spectrum of partially hyperbolic diffeomorphisms
本文研究部分双曲微分同胚的Lyapunov谱简单性，证明保体积类被简单谱开集累积，并给出基于同宿相关周期点的简单性判据。
We establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of Sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples.
We introduce random towers to study almost sure rates of correlation decay for random partially hyperbolic attractors. Using this framework, we obtain abstract results on almost sure exponential, stretched exponential and polynomial correlation decay rates. We then apply our results to small random perturbations of Axiom A attractors, small perturbations of derived from Anosov partially hyperbolic systems and to solenoidal attractors with random intermittency.
Non-hyperbolic ergodic measures and horseshoes in partially hyperbolic homoclinic classes
针对鲁棒非双曲传递微分同胚，本文证明其同宿类中每个遍历测度（含非双曲情形）均可由双曲集在弱*拓扑与熵意义下逼近。
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.
We prove the stochastic stability of an open class of partially hyperbolic diffeomorphisms, each of which admits two centers $E^c_1$ and $E^c_2$ such that any Gibbs $u$-state admits only positive (resp. negative) Lyapunov exponents along $E^c_1$ (resp. $E^c_2$).
Let M be a closed, symplectic connected Riemannian manifold, f a symplectomorphism on M. We prove that if f is C1-stably weakly shadowing on M, then the whole manifold M admits a partially hyperbolic splitting.
Partially hyperbolic lattice actions on 2-step nilmanifolds
本文研究高阶秩格在2步幂零流形上包含部分双曲元素的作用，在技术假设下证明其必为仿射映射，从而推广了Brown等人的刚性结果。
Entropy properties of mostly expanding partially hyperbolic diffeomorphisms
本文研究主要扩张的部分双曲微分同胚的熵性质，证明在极小强稳定叶状结构与一维中心丛条件下，其$C^1$邻域内拓扑熵连续且满足中间熵性质。
In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.
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