Xiongping Dai
Using a sifting-shadowing combination, we prove in this paper that an arbitrary $\mathrm{C}^1$-class local diffeomorphism $f$ of a closed manifold $M^n$ is uniformly expanding on the closure $\mathrm{Cl}_{M^n}(\mathrm{Per}(f))$ of its periodic point set $\mathrm{Per}(f)$, if it is nonuniformly expanding on $\mathrm{Per}(f)$.
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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。
This article is devoted to level-1 large deviation properties in some nonuniformly hyperbolic systems via Pesin theory. In particular, our result can be applied to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.
This article is devoted to the study of the historic set, which was introduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for historic sets. Our results can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.
We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.
For a surface diffeomorphism, a compact invariant locally maximal set $W$ and some subset $A\subset W$ we study the $A$-exceptional set, that is, the set of points whose orbits do not accumulate at $A$. We show that if the Hausdorff dimension of $A$ is smaller than the Hausdorff dimension $d$ of some ergodic hyperbolic measure, then the topological entropy of the exceptional set is at least the entropy of this measure and its Hausdorff dimension is at least $d$. Particular consequences occur when there is some a priori defined hyperbolic structure on $W$ and, for example, if there exists an SR…
For a large class of nonuniformly expanding maps of $\Bbb R^m$, with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques previously used for expanding maps on quasi-Hölder spaces. We give general conditions and provide examples to which apply our result.
We estimate the speed of decay of correlations for general nonuniformly expanding dynamical systems, using estimates on the time the system takes to become really expanding. Our method can deal with fast decays, such as exponential or stretched exponential. We prove in particular that the correlations of the Alves-Viana map decay in $O(e^{-c \sqrt{n}})$.
This article is devoted to the investigation of the topological pressure of generic points for nonuniformly hyperbolic systems via Pesin theory. In particular, our result can be applied to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.
We give a unified treatment of decay of correlations for nonuniformly expanding systems with a good inducing scheme. In addition to being more elementary than previous treatments, our results hold for general integrable return time functions under fairly mild conditions on the inducing scheme.
For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.
Exact dimensionality of stationary measures for random diffeomorphisms with all Lyapunov exponents equal and negative
本文证明在单一负李雅普诺夫指数条件下，非均匀共形收缩随机微分同胚的平稳测度具有精确维数，其维数由Furstenberg熵与负指数之比给出，且无需离散性假设。
For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which exhibit nonuniformly hyperbolic behavior. For these parameters, we prove the existence of absolutely continuous invariant measures with good statistical properties, such as exponential decay of correlations. Combining our construction with the logarithmic nature of the singularities, we obtain a positive variance in Central Limit Theorem, for any nonconstant Hö…
We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate $C^*\cdot\frac{\ln m}{m}$, where $C^*$ is a computable fixed constant and $m^{-1}$ is the mesh size of the discretization.
Schmidt's Game and Nonuniformly Expanding Interval Maps
研究Manneville-Pomeau映射，证明前向轨道避开左端点为0的区间的点集在Schmidt博弈中强获胜，该集具有稠密性、满Hausdorff维数和可数交性质。
The transfer operator corresponding to a uniformly expanding map enjoys good spectral properties. Here it is verified that coupling yields explicit estimates that depend continuously on the expansion and distortion constants of the map. For nonuniformly expanding maps with a uniformly expanding induced map, we obtain explicit estimates for mixing rates (exponential, stretched exponential, polynomial) that again depend continuously on the constants for the induced map together with data associated to the inducing time. Finally, for nonuniformly hyperbolic transformations, we obtain the correspo…
Horseshoes for a class of nonuniformly expanding random dynamical systems on the circle
本文为一维随机动力系统引入随机马蹄概念，并证明在加性噪声足够大且李雅普诺夫指数为正时，一类圆周自同态中随机马蹄大量存在，其诱导动力学可实现两个符号上的全移位。
We prove that the statistical properties of random perturbations of a nonuniformly hyperbolic diffeomorphism are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain $C^2$-open class of nonuniformly hyperbolic diffeomorphisms introduced in [Alves, J; Bonatti, C. and Viana, V., SRB measures for partially hyperbolic systems with mostly expanding central direction, Invent. Math., 140 (2000), 351-398] are stochastically stable. Our setting encompasses that of partially hy…
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
Let $f$ be a $C^{1+α}$ nonuniformly hyperbolic diffeomorphism. We use a a nonadditive version of the topological pressure of a class of admissible, possibly noncontinuous potentials $P^*(Φ)$ to prove the following variational equation: $P^*(Φ) = \sup_{Ω\in {\mathcal H}}P^*(f|Ω,Φ)$ supremum taken over the set ${\mathcal H}$ of basic subsets in $M$. As a consequence we find a lower bound for the Cantor dimension of the stable and unstable Cantor sets of a non trivial conformal nonuniformly hyperbolic isolated sets.
Mostly nonuniformly sectional expanding systems
本文提出几乎非均匀横截扩张系统概念，展示了一个非均匀横截超几何集的例子，该集满足MNUSE但不满足横截超几何或渐近横截性，并给出了高共维非均匀横截扩张吸引子的构造。
In this paper we study some skew product diffeomorphisms with nonuniformly hyperbolic structure along fibers. We show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers.
We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in whi…
In this paper we prove that for an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $ω$-full measured set $\widetildeΛ$ such that for every invariant probability $μ\in \mathcal{M}_{inv}(\widetildeΛ,f)$, the metric entropy of $μ$ is equal to the topological entropy of saturated set $G_μ$ consisting of generic points of $μ$: $$h_μ(f)=h_{\top}(f,G_μ).$$ Moreover, for every nonempty, compact and connected subset $K$ of $\mathcal{M}_{inv}(\widetildeΛ,f)$ with the same hyperbolic rate, we compute the topological entropy of saturated set $G_K$…
Mixing properties of a class of nonuniformly expanding maps. Application to H{ö}lderian invariance principles
本文研究返回时间仅具有p>1阶弱矩的非一致扩张映射的混合性质，并由此建立Hölder连续观测量部分和过程的Hölder不变原理，进而将结果推广到间歇映射及BV观测量。
We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.
This is a significantly expanded version of the survey paper "Mixing and decay of correlations in non-uniformly expanding maps: a survey of recent results" math/0301319. We discuss recent results on decay of correlations for non-uniformly expanding maps. Throughout the discussion, we address the question of why different dynamical systems have different rates of decay of correlations and how this may reflect underlying geometrical characteristics of the system. We also give detailed sketches of some results such as the existence of absolutely continuous invariant measures for Markov expanding …
Gouezel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.
Let $f$ be a $C^{1+α}$ diffeomorphism of a compact Riemannian manifold and $μ$ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential $ϕ$ there exists a sequence of basic sets $Ω_n$ such that the topological pressure $P(f|Ω_n,ϕ)$ converges to the free energy $P_μ(ϕ) = h(μ) + \intϕ{dμ}$. Then we introduce a class of potentials $ϕ$ for which there exists sequence of basic sets $Ω_n$ such that $P(f|Ω_n,ϕ) \to P(ϕ)$.
Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $μ$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, μ)$. In smooth ergodic theory there are various natural choices of $μ$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We sh…
Quasi-shadowing property for nonuniformly partially hyperbolic systems
本文研究非一致部分双曲系统，建立自适应伪轨道运动的拟跟踪性质，并考察拟 specification 与拟闭合性质，进而将 Katok 周期轨道增长结果推广到任意遍历测度。
We study skew-products of the form (x,ω)\mapsto (Tx, ω+ϕ(x)) where T is a nonuniformly expanding map on a space X, preserving a (possibly singular) probability measure \tildeμ, and ϕ:X\to S^1 is a C^1 function. Under mild assumptions on \tildeμand ϕ, we prove that such a map is exponentially mixing, and satisfies the central and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi.
Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $μ$ with positive entropy, and let $\mathcal{A}$ be a Hölder continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mat…
We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where the decay rate $O(1/n^β)$ has previously been proved for diffeomorphisms, we establish the decay rate $O(1/t^β)$ for typical flows. Applications include certain classes of semidispersing billiards, as well as dispersing billiards with vanishing curvature. In addition, we obtain results for suspension…
We consider deterministic homogenization (convergence to a stochastic differential equation) for multiscale systems of the form \[ x_{k+1} = x_k + n^{-1} a_n(x_k,y_k) + n^{-1/2} b_n(x_k,y_k), \quad y_{k+1} = T_n y_k, \] where the fast dynamics is given by a family $T_n$ of nonuniformly expanding maps. Part 1 builds on our recent work on martingale approximations for families of nonuniformly expanding maps. We prove an iterated weak invariance principle and establish optimal iterated moment bounds for such maps. (The iterated moment bounds are new even for a fixed nonuniformly expanding map T.)…
We prove results on mixing and mixing rates for toral extensions of nonuniformly expanding maps with subexponential decay of correlations. Both the finite and infinite measure settings are considered. Under a Dolgopyat-type condition on nonexistence of approximate eigenfunctions, we prove that existing results for (possibly nonMarkovian) nonuniformly expanding maps hold also for their toral extensions.
Stable manifolds, Horseshoes and Lyapunov exponents for $C^1$ diffeomorphisms without domination
本文提出了一种针对$C^1$微分同胚的非均匀超双曲理论，通过引入新的共振块，研究稳定流形、蛇形结构及李亚普诺夫指数的性质。
For an ergodic hyperbolic measure $ω$ of a $C^{1+α}$ diffeomorphism, there is an $ω$ full-measured set $\tildeΛ$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tildeΛ)$ coincides with the accumulating set of time averages of Dirac measures supported at {\it one orbit}, where $\mathbb{M}_{inv}(\tildeΛ)$ denotes the space of invariant measures supported on $\tildeΛ$. Such state points corresponding to a fixed $V$ are dense in the support $supp(ω)$. Moreover, $\mathbb{M}_{inv}(\tildeΛ)$ can be accumulated by time averages of Dirac measures supported at {\it one o…
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…
We prove that a structurally stable diffeomorphism of a closed (2m+1)-manifold has no codimension one non-orientable expanding attractors.
We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any $C^{1+}$ partially volume expanding diffeomorphisms admits finitely many physical measures, the union of whose basins has full volume.
We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state for a large class of non- uniformly expanding transformations and Holder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_θ$ of functions with essential bounded variation that strictly contain Holder con…
In this work we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that the mostly expanding condition guarantee the existence of physical measures and provide more information about the statistics of the system. Mañé's classical derived-from-Anosov diffeomorphism on $\mathbb{T}^3$ belongs to this set.
We discuss dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robust topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.
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。
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. $E^c_2$).
On isolated periodic points of diffeomorphisms with expanding attractors of codimension 1
研究$Ω$-稳定3-微分同胚中孤立周期点与余维1扩张吸引子的关系，利用吸引子结构估计孤立周期点的最小数量。
In this paper we study physical measures for $\C^{1+α}$ partially hyperbolic diffeomorphisms with mostly expanding center. We show that every diffeomorphism with mostly expanding center direction exhibits a geometrical-combinatorial structure, which we call skeleton, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how physical measures bifurcate as the diffeomorphism changes under $C^1$ topology. Moreover, for each diffeomorphism with mostly expanding center, there exists a $C^1$ neighborhood, such that diffeomorphism am…
For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for which there exist no heteroclinic intersections. We build the one-to-one corresponding between periodic points in any skeleton and physical measures. By making perturbations on skeletons, we study the continuity of physical measures with respect to dynamics under $C^1$-topology.
SRB measures for mostly expanding partially hyperbolic diffeomorphisms via the variational approach
本研究采用变分方法，证明了主要扩张的部分双曲$\u2102^1$微分同胚及一类控制分裂情形下SRB测度的存在性，并在$\u2102^{1+α}$正则性下验证了盆覆盖性质。
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…
The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another…
J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization of nonuniform hyperbolicity and show that it is equivalent to invertibility of the same operator on a larger, Fréchet space. We apply it to obtain a condition for a diffeomorphism on the boundary of the set of Anosov diffeomorphisms to be nonuniformly hyperbolic. Finally we generalise the Shadowing lemma in the same context.
We prove the almost sure invariance principle with rate $o(n^{\varepsilon})$ for every $\varepsilon > 0$ for Hölder continuous observables on nonuniformly expanding and nonuniformly hyperbolic transformations with exponential tails. Examples include Gibbs-Markov maps with big images, Axiom A diffeomorphisms, dispersing billiards and a class of logistic and Hénon maps. The best previously proved rate is $O(n^{1/4} (\log n)^{1/2} (\log \log n)^{1/4})$. As a part of our method, we show that nonuniformly expanding transformations are factors of Markov shifts with simple structure and natural metri…
We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.
We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive, such a diffeomorphism has a unique physical measure that persists and varies continuously with the dynamics. A main ingredient in the proof is a new Pliss-like Lemma which, under the right circumstances, yields frequency of hyperbolic times close to one. Another novelty is the introduction of a new characterization of Gibbs $cu$-states. Both of these may be of…
In the present paper, we study the distribution of the return points in the fibers for a RDS (random dynamical systems) nonuniformly expanding preserving an ergodic probability, we also show the abundance of nonlacunarity of hyperbolic times that are obtained along the orbits through the fibers. We conclude that any ergodic measure with positive Lyapunov exponents satisfies the nonuniform specification property between fibers. As consequences, we prove that any expanding measure is the limit of probability measure whose measures of disintegration on the fibers are supported by a finite number …
We prove statistical limit laws for sequences of Birkhoff sums of the type $\sum_{j=0}^{n-1}v_n\circ T_n^j$ where $T_n$ is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family $T_n$ is replaced by a fixed transformation $T$, and which is particularly effective in the case when $T_n$ varies with $n$. In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family $T_n$ consists of intermittent …
Expanding on average diffeomorphisms of surfaces: exponential mixing
本文证明了闭曲面上保体积微分同胚的平均扩张元组相关的 Bernoulli 随机动力系统具有指数混合性质。
Maximal transverse measures of expanding foliations
本研究针对微分同胚的膨胀叶状结构，通过动力学平均构造极大横截测度，证明其对一类微分同胚呈指数收敛且极限测度有有限遍历分解，揭示了该测度与极大u-熵测度的关联。
We establish two results under which the topology of a hyperbolic set constrains ambient dynamics. First if a set is a compact, transitive, expanding hyperbolic attractor of codimension 1 for some diffeomorphism, then it is a union of transitive, expanding attractors (or contracting repellers) of codimension 1 for any diffeomorphism such that it is hyperbolic. Secondly, if a set is a nonwandering, locally maximal, compact hyperbolic set for a surface diffeomorphism, then it is locally maximal for any diffeomorphism for which it is hyperbolic.
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