Ali Tahzibi
In this survey we recall basic notions of disintegration of measures and entropy along unstable laminations. We review some roles of unstable entropy in smooth ergodic theory including the so-called invariance principle, Margulis construction of measures of maximal entropy, physical measures and rigidity. We also give some new examples and pose some open questions.
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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。
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.
This is an expository article/encyclopedia entry explaining the history, techniques, and central results in the field of smooth ergodic theory.
In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was done in the series of papers [BL1-BL5], but the approach to the general smooth case must be different.
We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$.
In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{Hölder}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action and establish controls on their local geometry. In the second part we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier--You…
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.
Homoclinic classes for flows: ergodicity and SRB measures
本文研究了流中同宿类的遍历性与SRB测度，证明了特定条件下同宿类交集的遍历性，并探讨了正则SRB测度的相关结果。
For any $C^1$ diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach to this result is under the settings of topological dynamical systems, which is also applicable to infinite dimensional $C^1$ dynamical systems.
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.
We augment the method of $C^\infty$ conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.
In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable holonomy. We prove that partial hyperbolic attractors satisfying conditions of transversality between unstable leaves via the stable holonomy, neutrality in the central direction and regularity of the stable foliation admits a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. More…
Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.
The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group $\H$. We establish the decomposition of the tangent space of any $C^\infty$ compact Riemannian manifold $M$ for Lyapunov exponents, and show that all Lyapunov exponents for the center elements are zero. We obtain that if an $\H$ group action contains an Anosov element, then under certain conditions on the element, the center elements are of finite order. In particular there is no faithful codimensional one Anosov Heisenberg group action on any manifolds, and no faithful codimen…
Entropies and volume growth of unstable manifolds
本文通过将不稳定叶层熵与覆盖数关联，给出了紧流形上 $C^2$ 微分同胚遍历测度度量熵的新上界，并推广至 $C^{1+α}$ 情形。
This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carathéodory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free ener…
It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.
A measured solenoid is a laminated space endowed with a tranversal measure invariant by holonomy, as defined in arXiv:0910.2836. A measured solenoid immersed in a smooth manifold produces a closed current (known as generalized Ruelle-Sullivan current). Uniquely ergodic solenoids are those for which there is a unique (up to scalars) transversal measure. By the results in arXiv:0910.2913, for any smooth manifold, any real homology class is represented by a uniquely ergodic solenoid. In this paper, we prove that the currents associated to uniquely ergodic solenoids are dense in the space of close…
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. I…
Local stable and unstable sets for random dynamical systems
针对正熵双侧连续丛随机动力系统，在缺乏光滑性时用局部稳定/不稳定集替代不变流形，证明纤维熵由不稳定集上的作用决定，并建立其Hausdorff维数下界。
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。
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。
In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $μ$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(μ)$ of periodic orbits for the homoclinic relation, called its \e…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds.
Maximal Entropy Measures for Non-Accessible Topological Skew Products
本文研究一维紧致中心叶层的部分双曲微分同胚，证明在流形为不稳定叶状结构饱和的唯一极小不变集时，极大熵测度要么唯一且非双曲，要么恰有两个双曲遍历测度。
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental propert…
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.
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.
For a non-conformal repeller $Λ$ of a $C^{1+α}$ map $f$ preserving an ergodic measure $μ$ of positive entropy, this paper shows that the Lyapunov dimension of $μ$ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+α}$ diffeomorphism $f$ preserving a hyperbolic ergodic measure $μ$ of positive entropy, if $(f, μ)$ has only two Lyapunov exponents $λ_u(μ)>0>λ_s(μ)$, then the Hausdorff or lower box or upper box dimension of $μ$ can be approximated by the corresponding dimension of the horseshoes $\{Λ_n\}$. The same statement holds true if $…
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.
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 classify quasiconformal Anosov flows whose strong stable and unstable distributions are at least two dimensional and the sum of these two distributions is smooth. We deduce from this classification result the complete classification of volume-preserving quasiconformal diffeomorphisms whose stable and unstable distributions are at least two dimensional. Our central idea is to take a good time change so that perodic orbits are equi-distributed with respect to a lebesgue 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。
Smooth center-stable/unstable manifolds and foliations of stochastic evolution equations with non-dense domain
本文利用积分半群理论和随机Stieltjes卷积估计，通过Lyapunov-Perron方法研究了一类非稠密域随机偏微分方程的中心-不稳定不变流形与中心-稳定叶层的存在性和光滑性，并用随机年龄结构模型和随机抛物型方程加以验证。
We define a general notion of a smooth invariant (central) ergodic measure on the space of paths of an $N$-graded graph (Bratteli diagram). It is based on the notion of standardness of the tail filtration in the space of paths, and the smoothness criterion uses the so-called intrinsic metric which can be canonically defined on the set of vertices of these graphs. In many cases known to the author, like the Pascal graph, the Young graph, the space of configurations, all ergodic central measures are smooth (in this case, we say that the graph is smooth). But even in these cases, the intrinsic me…
Dense Phenomena for Ergodic Schrödinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
本文证明对紧致度量空间上同胚的遍历轨道采样得到的薛定谔算子，存在稠密采样函数集使谱间隙有限、态密度积分光滑且李雅普诺夫指数光滑为正，并肯定回答Walters的非一致上循环存在性问题。
In this note we consider autonomous SDEs admitting smooth invariant measures. We present a method in finding (almost everywhere) good bounds for $\sup \{\|X_t\|: t \in [0, T]\}$ for strong solutions $X_{\cdot}$ to such SDEs, which in many cases are optimal bounds. In some situation (especially in one-dimensional SDEs' cases), the discarded measure-zero set can be chosen to be a measure-zero set of the underlying Brownian motion uniform for all initial points $X_0=x$.
We show that the ergodic averages for the horocycle flow on the two-torus associated by Giulietti and Liverani to an Anosov diffeomorphism either grow linearly or are bounded, in other words there are no deviations. For this, we use topological invariance of the Artin-Mazur zeta function to exclude resonances outside of the open unit disc. Transfer operators acting on suitable spaces of anisotropic distributions and their Ruelle determinants are the key tools in the proof. As a bonus, we show that for any smooth Anosov diffeomorphism F on the two-torus, the correlations for the measure of maxi…
We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.
For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
Propagation of regularity along unstable manifolds
该研究证明了在不稳定流形上光滑的共振态在整个流形上也是光滑的，并展示了叶状空间伪微分算子理论的应用。
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