Huyi Hu, Anna Talitskaya
We construct a smooth hyperbolic volume preserving diffeomorphism on a four dimensional compact Riemannian manifold which has countably many ergodic components and is arbitrarily close to the identity map.
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We prove that in the isotopy class of any volume preserving partially hyperbolic diffeomorphism in a $3$-dimensional manifold, there is a non-partially hyperbolic stably ergodic diffeomorphism. In particular, we provide new examples of stably ergodic diffeomorphisms in 3-dimensional manifolds with respect to a smooth volume measure.
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.
In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the $C^1$-density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms.
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.
We study the behavior of diffeomorphisms, contained in the closure $\bar {\A_\a}$ (in the inductive limit topology) of the set $\A_\a$ of real-analytic diffeomorphisms of the torus $\Bbb T^2$, conjugated to the rotation $R_\a:(x,y)\mapsto (x + \a, y)$ by an analytic measure-preserving transformation. We show that for a generic $\a\in [0,1]$, $\bar {\A_\a}$ contains a dense set of uniquely ergodic diffeomorphisms. We also prove that $\bar {\A_\a}$ contains a dense set of diffeomorphisms that are minimal and non-ergodic.
We prove that for every $ε>0$ there exists a minimal diffeomorphism $f:\T^{2}\rightarrow\T^{2}$ of class $C^{3-ε}$ and semiconjugate to an ergodic traslation, and have the following properties: zero entropy, sensitivity with respect to initial conditions and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé's example of derived from Anosov diffeomorphism on $\T^3.$
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.
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.
Characterization of Ergodic Measures of Maximal Entropy for Topologically Transitive Partially Hyperbolic Diffeomorphisms with compact center leaves
该文研究三维环面上具紧一维中心叶的拓扑传递部分双曲微分同胚，给出零Lyapunov指数的最大熵遍历测度的数量上界，并完整刻画其支撑集。
In this note we prove that a homeomorphism is countably-expansive if and only if it is measure-expansive. This result is applied for showing that the $C^1$-interior of the sets of expansive, measure-expansive and continuum-wise expansive $C^1$-diffeomorphisms coincide.
In this paper, we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on $3-$torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so-called Kan-type diffeomorphisms admitting physical measures with intermingled basins.
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。
In this work we present an example of C^\infty-diffeomorphism of a compact 4-manifold such that it admits a global SRB measure μbut for which the special ergodic theorem doesn't hold. Namely, for this transformation there exist a continuous function ϕand a positive constant αsuch that the following holds: the set of the initial points for which the Birkhoff time averages of the function ϕdiffer from its μ--space average by at least αhas zero Lebesgue measure but full Hausdorff dimension.
In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center…
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.
Let $ G $ be a connected, simply connected nilpotent Lie group and $ Γ< G $ a lattice. We prove that each ergodic diffeomorphism $ ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $ G/Γ$, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(Γ)=Γ$, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short int…
We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the w…
Ergodic measures with large entropy have long unstable manifolds for $C^\infty$ surface diffeomorphisms
该研究针对 $C^\infty$ 曲面微分同胚，证明了具有大度量熵的遍历测度对应点集的不稳定流形普遍较长，给出了熵下界与不稳定流形尺寸、测度下界间的量化关系，丰富了动力系统遍历理论。
A volume preserving nonuniformly hyperbolic diffeomorphism with arbitrary number of ergodic components and close to the identity
本文证明任意维数至少为5的紧致光滑黎曼流形上存在保持体积的非一致双曲微分同胚，其遍历分量个数可任意指定（包括无穷），且可任意接近恒等映射。
If a non-ergodic, partially hyperbolic diffeomorphism on the 3-torus is homotopic to an Anosov diffeomorphism $A$, it is topologically conjugate to $A$.
We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.
Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms
本文研究具有一维中心方向的部分双曲微分同胚，在极小叶状结构和blender-horseshoes假设下证明了熵与中心Lyapunov指数的联合灵活性，并给出广泛适用例子。
A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].
We give a description of ergodic components of SRB measures in terms of ergodic homoclinic classes associated to hyperbolic periodic points. For transitive surface diffeomorphisms, we prove that there exists at most one SRB measure.
In many dynamical systems, countably infinitely many invariant tori co-exist. The occurrence of quasiperiodicity on any one of these tori is sometimes sufficient to establish strong global properties, like dense trajectories and periodic points. In this paper, we establish sufficient conditions for a countably infinite collection of parameterized circle diffeomorphisms to have quasiperiodic behavior on at least one of the circles, for a full Lebesgue measure set of the parameter values. As an application, we study parameterized families of skew-product maps on the torus and prove sufficient co…
We show that every codimension one partially hyperbolic diffeomorphism must support on $\mathbb{T}^{n}$. It is locally uniquely integrable and derived from a linear codimension one Anosov diffeomorphism. Moreover, this system is intrinsically ergodic, and the A. Katok's conjecture about the existence of ergodic measures with intermediate entropies holds for it.
Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms
本文研究了$C^r$表面微分同胚最大熵遍历测度的连续性性质，证明了其数量在特定条件下上半连续，并揭示了局部常数性的条件及极限点的遍历性。
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.
In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This …
Intrinsic Ergodicity, Generators and Symbolic Representations of Algebraic Group Actions
本文为可数无限顺从群的内禀遍历主代数作用构造自然符号表示，并据此找到显式生成划分。
We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.
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。
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 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.
In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems including singular hyperbolic attractors, $C^\infty$ surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.
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 give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a $C^1$-dense and open subset of the set of a diffeomorphisms having a robust cycle. As a corollary, there exists a $C^1$-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting…
In this paper, we define and study unstable measure theoretic pressure for C^1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. For any ergodic measure, we show that this unstable metric pressure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponents with respect to the ergodic measure. On the other hand, we also define unstable topological and metric pressure in terms of the Bowen's picture and the capacity picture. We show that all these definitions of unstable metric pressure actually coincide for any ergodic measure.
A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures
在保守部分双曲三维环境中，针对三类微分同胚建立双曲性与非双曲遍历测度的二分法，方法无需扰动，还独立于保守假设构造了非双曲遍历测度，可应用于更多情境。
We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.
We construct examples of volume-preserving uniquely ergodic (and hence minimal) real-analytic diffeomorphisms on odd-dimemsional spheres
We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.
We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area--preserving $C^2$--diffeomorphism with polynomial uniform growth of the derivative is $C^2$--conjugate to a 2--steps skew product of the form \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+α,\ep x_2+β(x_1),x_3+γ(x_1,x_2))\in\tor^3,\] where $\ep=\pm 1$. We also indicate why there is no 4--dimensional analogue of the above result. Ra…
In this paper, we study ergodic properties of hyperbolic measures with local product structure. We show that all the classical results that hold in the case of SRB measure hold for these measures. In particular, we show the decomposition in countably many ergodic components, we prove the decomposition into K-components, and show that for hyperbolic measure with local product structure, The K property implies the Bernoulli property. We also give some examples of measures where the results are applicable.
We consider diffeomorphisms in the $C^\infty$-closure of the conjugancy class of translations of the 2-torus. By a theorem of Fathi and Herman, a generic diffeomorphism in that space is minimal and uniquely ergodic. We define a new mixing-like property, which takes into account the "directions" of mixing, and we prove that generic elements of the space in question satisfy this property. As a consequence, we show that there is a residual set of strictly ergodic diffeomorphisms without invariant foliations of any kind. We also obtain an analytic version of these results.
We show that a stably ergodic diffeomorphism can be $C^1$ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.
Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.
We consider the space of $C^1$-diffeomorphims equipped with the $C^1$-topology on a three dimensional closed manifold. It is known that there are open sets in which $C^1$-generic diffeomorphisms display uncountably many chain recurrences classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are $C^1$-generic diffeomorph…
We consider C1 Anosov diffeomorphisms on a compact Riemannian manifold. We define the weak pseudo-physical measures, which include the physical measures when these latter exist. We prove that ergodic weak pseudo-physical measures do exist, and that the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak*-closed convex hull of the ergodic weak pseudo-physical measures. In brief, we give in the C1-scenario of uniform hyperbolicity, a characterization of Pesin's Entropy Formula in terms of physical-like properties.
In this paper, we provide a technique result on the existence of Gibbs cu-states for diffeomorphisms with dominated splittings. More precisely, for given $C^2$ diffeomorphim $f$ with dominated splitting $T_ΛM = E\oplus F$ on an attractor $Λ$, by considering some suitable random perturbation of $f$, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $E$, then its ergodic components contain Gibbs cu-states associated to $E$. With this technique, we show the existence of SRB measures and physical measur…
We construct a plethora of Anosov-Katok diffeomorphisms with non-ergodic generic measures and various other mixing and topological properties. We also construct an explicit collection of the set containing the generic points of the system with interesting values of its Hausdorff dimension.
Physical measures on partially hyperbolic diffeomorphisms with multi 1-D centers
该研究针对所有Gibbs u-态为双曲的多一维中心部分双曲微分同胚，证明遍历物理测度有限性，建立盆覆盖性质判定准则，给出该性质成立的Lyapunov指数符号条件。
The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some classes of transformations that have special properties, such as the collection of transformations with discrete spectrum or Bernoulli shifts. This paper shows that a general classification is impossible (even in concrete settings) by showing that the collection $E$ of pairs of ergodic, Lebesgue measure preserving diffeomorphisms $(S,T)$ of the 2-torus that are is…
Finite Measures of Maximal Entropy for an Open Set of Partially Hyperbolic Diffeomorphisms
本文研究一维中心方向且不稳定熵大于稳定熵的部分双曲微分同胚，证明其仅有有限个遍历最大熵测度，并将相关结果推广至任意维度。
We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $μ$ which has positive entropy and is such that for a continuous function $ϕ\colon X\to\mathbb{R}$ the integral $\intϕ\,dμ$ vanishes. The criterion is an extended version of …
In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results …
This article is devoted to the study of the multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps.
On the cardinality of measures of maximal relative entropy for smooth skew products
本文针对传递Anosov基上的$\mathcal C^{1+\alpha}$光滑斜积微分同胚，证明其最大相对熵遍历双曲测度至多可数，且二维纤维情形下正相对拓扑熵时该类遍历测度也至多可数。
Discontinuity of Lyapunov exponents vs Entropy for smooth surface diffeomorphisms
研究光滑曲面微分同胚中Lyapunov指数与熵的关系，通过简化论证展示最大熵测度附近指数的连续性性质。
We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of points with only non-zero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independe…
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