Flavio Abdenur, Sylvain Crovisier
We prove that, on connected compact manifolds, both C1-generic conservative diffeomorphisms and C1-generic transitive diffeomorphisms are topologically mixing. This is obtained through a description of the periods of a homoclinic class and by a control of the period of the periodic points given by the closing lemma.
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We prove transitivity for volume preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^3$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.
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 .
It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Under suitable hypothesis on Lyapunov exponents on the set of periodic points and the structure of the MME we get transitivity of $C^1-$Anosov diffeomorphisms.
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.
Topological transitivity of Kan-type partially hyperbolic diffeomorphisms
本文证明了加厚环面上包括Kan 1994年部分双曲例子在内的一类微分同胚具有拓扑传递性，该例子以混合盆现象著称。
On the Topological Transitivity of Interval Extensions of Anosov Diffeomorphisms
本文证明了加厚幂零流形上一类解析部分双曲微分同胚（由Anosov微分同胚的区间扩张构造的斜积系统）的拓扑传递性，并提出了允许中心方向任意有限阶抛物退化的机制。
We prove that the action of the semigroup generated by a $C^r$ generic pair of area-preserving diffeomorphisms of a compact orientable surface is transitive.
Given a $C^2$- Anosov diffemorphism $f: M \rightarrow M,$ we prove that the jacobian condition $Jf^n(p) = 1,$ for every point $p$ such that $f^n(p) = p,$ implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state a classical theorem of Livsic-Sinai without directly assuming transitivity as a general hypothesis. A special consequence of our result is that every $C^2$-Anosov diffeomorphism, for which every point is regular, is indeed transitive.
Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms
研究部分双曲微分同胚，通过仔细研究极小\(\mathcal{W}^u -\)饱和集，证明了在一维中心的部分双曲微分同胚中有\(C^1 -\)开且稠密子集，存在恰当拟吸引子或两个强叶状结构是极小的，保体积情形同理且有保守外的鲁棒传递性结果。
We consider extensions of Anosov diffeomorphisms of an infranilmanifold by the real vector space R^ω. Our main result, based on the analogous theorem in finite dimensions proven by Nitica and Pollicott, is that any Holder cocycle satisfying an obvious obstruction induces a topologically transitive extension (topologically weak mixing, in fact). We show how to construct cocycles satisfying these conditions for any Anosov diffeomorphism, and then observe that unlike the finite dimensional case, where cocycles satisfying the obstruction are C^0-stably transitive, there can be no infinite-dimensio…
In this paper, we provide a new criterion for the stable transitivity of volume preserving finite generated group on any compact Riemannian manifold. As one of our applications, we generalised a result of Dolgopyat and Krikorian in \cite{DK} and obtained stable transitivity for random rotations on the sphere in any dimension. As another application, we showed that for $\infty \geq r \geq 2$, any $C^r$ volume preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer $K$…
Robust Transitivity of Partially Hyperbolic Diffeomorphisms with Interval Central Leaves
研究具有区间中心叶的边界保持部分双曲微分同胚，通过边界互连刻画其$C^k$鲁棒传递性，并将其应用于边界SRB测度的鲁棒混合盆现象刻画。
We prove that C^1-robustly transitive diffeomorphisms on surfaces with boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces with boundary which are C^k-robustly transitive, with k greater or equal than 2. This class of diffeomorphisms are examples where a version of Palis' conjecture on surfaces with boundary, about homoclinic tangencies and uniform hyperbolicity, does not hold in the C^2-topology. This follows showing that blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become C^2-robustly topologically mixing diffeomorphisms on a surfaces with b…
Hyperbolicity, transitivity and the two-sided limit shadowing property
本文刻画了闭流形上具有双侧极限伪轨跟踪性质的微分同胚，证明其C^1内部及通有情形均为传递Anosov微分同胚，从而将Anosov传递性猜想归结为该性质与Anosov性质的等价性。
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 prove that every $C^2$ Anosov diffeomorphism in a compact and connected Riemannian manifold has a unique SRB and physical probability measure, whose basin of attraction covers Lebesgue almost every point in the manifold. Then, we use structural stability of Anosov diffeomorphisms to deduce that all $C^1$ Anosov diffeomorphisms on compact and connected Riemannian manifolds are transitive.
In this paper we show the relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension 2 robustly transitive systems are robustly ergodic. For the three dimensional case, we define it almost robust ergodicity and prove that generically robustly transitive systems are almost robustly ergodic, if the Lyapunov exponents are nonzero. We also show in higher dimensions, that under some conditions robust transitivity implies robust ergodicity.
We consider extensions of the notion of topological transitivity for a dynamical system $(X,f)$. In addition to chain transitivity, we define strong chain transitivity and vague transitivity. Associated with each there is a notion of mixing, defined by transitivity of the product system $(X \times X, f \times f)$. These extend the concept of weak mixing which is associated with topological transitivity. Using the barrier functions of Fathi and Pageault, we obtain for each of these extended notions a dichotomy result that a transitive system of each type either satisfies the corresponding mixin…
We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.
Several stronger forms of transitivity in non-autonomous discrete dynamical systems
本文在非自治离散动力系统中引入并研究了多种更强的传递性概念，证明了其迭代不变性，给出了充分必要条件，并通过反例揭示了与自治系统的差异，最后建立了等价性条件。
On sufficient conditions for the transitivity of homeomorphisms
本文推导了具有跟踪性质的同胚拓扑传递的充要条件（含稠密不变子集的重心性质），对比Anosov微分同胚的相关条件，描述$C^1$内部、讨论例子并给出应用，为动力学传递性研究提供新依据。
We introduce and study two properties of dynamical systems: topologically transitive and topologically mixing under the set-valued setting. We prove some implications of these two topological properties for set-valued functions and generalize some results from single-valued case to set-valued case. We also show that both properties of set-valued dynamical systems are equivalence for any compact intervals.
In this paper we introduce and study some stronger forms of transitivity like total transitivity, weakly mixing for maps on G-spaces. We obtain their relationship with the earlier defined notion of strongly mixing for maps on G-spaces. We also study G-minimal maps on G-spaces in detail.
$C^r$-Chain closing lemma for certain partially hyperbolic diffeomorphisms
本文证明了部分双曲微分同胚的 $C^r$-链闭包引理，并由此得出通有情形下周期点稠密及链传递蕴含传递性。
Let $(X, f)$ be a topological dynamical system and $\mathcal {F}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an $\mathcal {F}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}: f^{n}(x) \in U\}$, is in $\mathcal {F}$; the system $(X,f)$ is called $\mathcal {F}$-point transitive if there exists some $\mathcal {F}$-transitive point. In this paper, we first discuss the connection between $\mathcal {F}$-point transitivity and $\mathcal {F}$-tran…
We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $Λ$ a non-trivial $C^1$-robustly transitive set of a vector field $X$, there is a $C^1$-perturbation $Y$ of $X$ such that the continuation $Λ_Y$ of $Λ$ is a topologically mixing set for $Y$. In particular, robustly transitive flows become topologically mixing after $C^1$-perturbations. These results generalize a theorem by Bowen on the basic sets of generic Axiom A flows. We also show that the set of flows whose non-trivial homoclinic classes are topologically mixing is \emph{n…
We introduce and study the topological concepts of chain transitivity, mixing and chain mixing property for dynamical systems induced by uniform hyperspaces. These notions generalize the relevant concepts for metric spaces.
In this paper, we introduce and study notions of average chain transitivity, average chain mixing, total average chain transitivity and almost average shadowing property. We also discuss their interrelations.
Transitivity And Related Notions For Graph Induced Symbolic Systems
本文利用二维图的邻接矩阵刻画其诱导移位空间的传递性、弱混合及方向性等动力学性质，给出水平与垂直传递性的充要条件，并研究图的低复杂度乘积分解。
Shadowing property and transitivity of a set-valued map and its inverse limit
研究集值映射及其逆极限的跟踪性、传递性等性质，证明了满射上半连续集值映射与其逆极限的跟踪性等价，且广义逆极限上移位映射的传递性等性质能推出集值映射相应性质，还给出了具有跟踪性的集值映射相关性质的等价关系。
We show that robustly transitive endomorphisms of a closed manifolds must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.
We study a partially hyperbolic and topologically transitive local diffeomorphism $F$ that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set $Λ$ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of $F|_Λ$ contains a gap and hence gives rise to a first order phase transition. A major part of t…
Analytic weakly mixing diffeomorphisms on odd dimensional spheres
本文提出共轭逼近方案，构造奇数维球面上关于体积测度弱混合的实解析微分同胚。
A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity
本文研究连续映射的度量传递、混合和超循环，引入了δ-拓扑传递、δ-拓扑混合及统一从下界的δ-混合，并证明了UFB-δ-TM蕴含δ-TM蕴含δ-TT。在可分F空间中，提出了δ-超循环准则，并证明其蕴含δ-超循环，同时经典超循环准则对任意δ>0都蕴含δ准则。
We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence …
A topological dynamical system $(X,f)$ is said to be multi-transitive if for every $n\in\mathbb{N}$ the system $(X^{n}, f\times f^{2}\times \dotsb\times f^{n})$ is transitive. We introduce the concept of multi-transitivity with respect to a vector and show that multi-transitivity can be characterized by the hitting time sets of open sets, answering a question proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661--1672]. We also show that multi-transitive systems are Li-Yorke chaotic.
In this paper, we prove that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. In addition, we show that any such group must always contain an element of positive topological entropy. We also provide an elementary characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.
The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.
Anti-classification results for weakly mixing diffeomorphisms
该研究将遍历理论反分类结果推广至弱混合系统，证明其同构与Kakutani等价关系非Borel集，表明无法通过可计算不变量对这类系统分类，还针对特定微分同胚验证了结论。
Strong Positive recurrence for potential and exponential mixing of equilibrium states of surface diffeomorphisms
本文研究了具有正熵的表面微分同胚的大类势能的强正回复性，并建立了平衡态的指数衰减相关函数和有效内在遍历性。
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.
In this paper we will show that if a sequence of natural numbers satisfies a certain growth rate, then there is a weak mixing diffeomorphism on $\mathbb{T}^2$ that is uniformly rigid with respect to that sequence. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly defined conjugation maps and the constructions are done in the $C^{\infty}$-topology as well as in the real-analytic topology.
Multivariate stable mixing local limit theorem for intermittent diffeomorphisms and flows
该研究针对超扩散区域的动力系统证明了多元稳定混合局部极限定理，并将其应用于间歇映射、螺线管等的可观测量及流相关问题。
Strong positive recurrence and exponential mixing for diffeomorphisms
本研究引入闭流形上微分同胚的强正递归（SPR）性质，证明其可通过具谱隙的可数状态马尔可夫移位编码，推导最大熵测度的多种遍历性质，验证正熵C∞曲面微分同胚及更多低正则、高维情形满足SPR。
On Transitivities for Skew Products
本文研究了非紧纤维斜积的传递性性质，探讨了普遍性条件和超循环准则与拓扑传递、弱混合和混合动力学性质的关系。
We show that on any smooth compact connected manifold of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{\infty}$-diffeomorphisms which preserve both a smooth volume $ν$ and a measurable Riemannian metric is dense in $\mathcal{A}_α \left(M\right)= \overline{ \left\{h \circ S_α \circ h^{-1} : h \in \text{Diff}^{\infty}\left(M, ν\right) \right\}}^{C^{\infty}}$ for every Liouvillean number $α$. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly c…
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
We prove that a real analytic pseudo-rotation $f$ of the disc or the sphere is never topologically mixing. When the rotation number of $f$ is of Brjuno type, the latter follows from a KAM theorem of Rüssmann on the stability of real analytic elliptic fixed points. In the non-Brjuno case, we prove that a pseudo-rotation of class $C^k$, $k\geq 2$, is $C^{k-1}$-rigid using the simple observation, derived from Franks' Lemma on free discs, that a pseudo-rotation with small rotation number compared to its $C^1$ (or Hölder) norm must be close to Identity. From our result and a structure theorem by Fr…
Topological transitivity of group cellular automata is decidable
本文证明了任意有限群上的d维群细胞自动机的拓扑transitivity是可判定的，并由此得到若干相关动力学性质的可判定性。
We show that topological mixing, weak mixing and total transitivity are equivalent for coded systems. We provide an example of a mixing coded system which cannot be approximated by any increasing sequence of mixing shifts of finite type, has only periodic points of even period and each set of its generators consists of blocks of even length. We prove that such an example cannot be a synchronized system. We also show that a mixing coded systems has the strong property $P$.
We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robust…
Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action ${\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t$, preserving a smooth volume $μ$. For any Liouville number $\a$ we construct a sequence of area-preserving diffeomorphisms $H_n$ such that the sequence $H_n\circ S_\a\circ H_n^{-1}$ converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc $\DD^2=\{x^2+y^2\leq 1\}$ or the closed annulus $\AAA=\TT\tim…
We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition (we provide an example of a dynamical system on a dendrite with dense endpoints satisfying this assumption). We also show that it may happen that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It may also happen that dynamical system is transitive but there is a unique periodic point, which in fact …
In this paper, some characterizations about transitivity, mildly mixing property, $\mathbf{a}$-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh's extensions restricted on some invariant closed subsets of the space of all upper semi-continuous fuzzy sets with the level-wise metric are obtained. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and $\mathbf{a}$-transitive, equicontinuous, uniformly rigid) if and only if the Zadeh's extension is transitive (resp., mildly mixing, $\mathbf{a}$-transitive, equicontinuous,…
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。
Weak mixing behavior for the projectivized derivative extension
本文在光滑和解析范畴中构造了拓扑熵为零、其微分在切丛射影化中关于光滑测度弱混合的二维流形微分同胚，核心方法是定量版共轭逼近技术。
These lecture notes focus on a recent result of Mike Hochman: an arbitrary standard Borel system can be embedded into a mixing Markov with equal entropy, respecting all invariant probability measures, with two exceptions: those carried by periodic orbits and those with maximal entropy. We discuss the corresponding notions of almost Borel embedding and isomorphism and universality. The main part of this paper is devoted to a self-contained and detailed proof of Hochman's theorem. We then explain how Katok's horseshoe theorem can be used to analyze diffeomorphisms with "enough" measures that are…
Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism
本研究证明零熵混合自同构（包括五维环面上的光滑微分同胚）在同构及Kakutani等价意义下不可分类，并给出反例说明笛卡尔积同构不能推出因子同构。
Let $(X,G,Φ)$ be a dynamical system, where $X$ is compact Hausdorff space, and $G$ is a countable discrete group. We investigate shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property, fix some finite subset $S\subset G$. We prove that if $X$ is totally disconnected, then $Φ$ has $S$-shadowing property if and only if $(X,G,Φ)$ is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric space (may be not totally disconnected), we prove that if $Φ$ has $S$-sha…
Mixing speed and stability of SRB measures through optimal transportation
本文提出基于最优传输的新方法研究Anosov微分同胚SRB测度，通过构建沿稳定叶状结构传输的Wasserstein度量变体使映射成为收缩，推导出SRB测度在特定方向上的指数相关性衰减及定量统计稳定性。
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