Peyman Eslami, Carlangelo Liverani
We establish sharp bounds on the mixing rates of a class of two dimensional non-uniformly hyperbolic symplectic maps. This provides a primer on how to investigate such questions in a concrete example and, at the same time, it solves a controversy between previous rigorous results and numerical experiments.
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We prove that a $C^1-$generic symplectic diffeomorphism is either Anosov or the topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of the periodic points. We also prove that $C^1-$generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and finally we give examples of volume preserving diffeomorphisms which are not point of upper semicontinuity of entropy function in $C^1-$topology.
We show that a topologically mixing $C^\infty$ Anosov flow on a 3 dimensional compact manifold is exponential mixing with respect to any equilibrium measure with Holder potential.
We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
We show that the differential in positive equivariant symplectic homology or linearized contact homology vanishes for non-degenerate Reeb flows with a continuous invariant Lagrangian subbundle (e.g. Anosov Reeb flows). Several applications are given, including obstructions to the existence of these flows and abundance of periodic orbits for contact forms representing an Anosov contact structure.
We prove that toric symplectic manifolds admit Hamiltonian pseudo-rotations with a finite, and in a sense minimal, number of ergodic measures. The set of ergodic measures of these pseudo-rotations consists of the measure induced by the symplectic volume form and the Dirac measures supported at the fixed points of the torus action. Our construction relies on the conjugation method of Anosov and Katok.
Exponential decay of random correlations for random Anosov systems mixing on fibers
本文借助Birkhoff锥方法，证明了满足Anosov性与纤维拓扑混合的外力驱动曲面Anosov系统，其对应唯一随机SRB测度的Hölder可观测量淬灭随机关联具有指数衰减的统计特性。
In this paper, we study the complicated dynamics of Anosov systems driven by an external force in the context of geometric theory (an abundance of random periodic points and random horseshoes) and smooth ergodic theory (random periodic measures and random Liv\v sic Theorem).
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.
Symplectic Geometry of Anosov Flows in Dimension 3 and Bi-Contact Topology
本文从接触与辛几何角度刻画三维 Anosov 流，建立研究其动力学的几何拓扑框架，并给出双接触结构唯一性及基于 Reeb 流的 Anosov 性判据。
We prove that in an open and dense set, Symplectic linear cocycles over time one maps of Anosov flows, have positive Lyapunov exponents for SRB measures.
We study circle extensions of analytic Anosov maps on the two torus: these are examples of partially hyperbolic maps for which the qualitative ergodic theory is well understood. In this paper we investigate rates of mixing (for the SRB measure) and prove explicit lower bounds involving the topological pressure of two times the unstable Jacobian. In particular we study the case when the extension function ("roof function") is given by a random trigonometric polynomial.
A symplectic viewpoint on Anosov flows
该综述沿接触与辛几何视角梳理三维 Anosov 流的几何理论，呈现若干原创结果，并提出相关开放问题与猜想。
We classify five dimensional Anosov flows with smooth decomposition which are in addition transversely symplectic. Up to finite covers and a special time change, we find exectly the suspensions of symplectic hyperbolic automorphisms of four dimensional toris, and the geodesic flows of three dimensional hyperbolic manifolds.
Stretched-exponential mixing for surface semiflows and Anosov flows
本文证明曲面悬挂半流在屋顶函数 Lipschitz 且非局部常上同调时关于 SRB 测度拉伸指数混合，并推广至双曲斜积、吸引子及余维一拓扑混合 Anosov 流。
We prove that an Anosov flow with $\mathcal{C}^{1}$ stable bundle mixes exponentially whenever the stable and unstable bundles are not jointly integrable. This allows us to show that if a flow is sufficiently close to a volume-preserving Anosov flow and $\operatorname{dim} \mathbb{E}_s = 1$, $\operatorname{dim} \mathbb{E}_u \geq 2$ then the flow mixes exponentially whenever the stable and unstable bundles are not jointly integrable.This implies the existence of non-empty open sets of exponentially mixing Anosov flows. As part of the proof of this result we show that $\mathcal{C}^{1+}$ uniforml…
We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic.
Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension $n\geq3$ admitting a metric of negative curvature (not necessarily equal to $g$) endowed with a twisted symplectic structure defined by a closed 2-form on $M$. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by the standard kinetic energy Hamiltonian, and we consider a compact regular energy level $Σ_{k}:=H^{-1}(k)$ of $H$. Suppose $Σ_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if the Hamiltonian flow restricted to $Σ_{k}$ is Anosov with $C^{1}$ weak bundles th…
Let $M$ be a closed $3$-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing.
We consider symplectic cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the Hölder topology.
Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by the maximality of the Toledo invariant. Then we concentrate on the particular case G=SP(2n,R), and we show that the image of Gamma under any maximal representation is a discrete faithful realization of Gamma as a Kleinian group of complex motions in X with an associated Anosov s…
There is a $C^1$-residual (Baire second class) subset $\mathcal{R}$ of symplectic diffeomorphisms on $2d$-dimensional manifold, $d\geq 1$, such that for every non-Anosov $f$ in $\mathcal{R}$ its topological entropy is lower bounded by the supremum of the Lyapunov exponents of their hyperbolic periodic points in the \emph{unbreakable central subbundle} (i.e., central direction with no dominated splitting) of $f$. The previous result deals with the fact that for $f$ in a residual set $\tilde{\mathcal{R}}$ of symplectic diffeomorphisms (containing $\mathcal{R}$) satisfies a trichotomy: or $f$ is …
Infinite ECH Capacities and Anosov Flows
本文建立了嵌入接触同调(ECH)与Anosov流动力学之间的联系，证明了许多辛4-流形的ECH容量无限，并给出了Reeb Anosov和Hamilton Anosov流存在的Floer理论障碍，应用于辛4-流形中亏格≥2的嵌入Lagrangian的存在性约束。
We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundle…
In the hyperspace of subcontinua of a compact surface we consider a second order Hausdorff distance. This metric space is compactified in such a way that the stable foliation of a pseudo-Anosov map is naturally identified with a hypercontinuum. We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric. Some dynamical properties of pseudo-Anosov maps, as topological mixing and the density of stable leaves, are generalized for cw-expansive homeomorphisms of pseudo-Anosov type on compact metric spaces.
Gibbs measures for contact Anosov flows are all exponentially mixing
通过增强Dolgopyat方法，证明接触Anosov流的所有吉布斯测度对Hölder观测具有指数衰减相关性。
On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the …
In the late seventies, Sullivan showed that for a convex cocompact subgroup $Γ$ of $\operatorname{SO}^\circ(n,1)$ with critical exponent $δ>0$, any $Γ$-conformal measure on $\partial \mathbb{H}^n$ of dimension $δ$ is necessarily supported on the limit set $Λ$ and that the conformal measure of dimension $δ$ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$ of rank at most $3$. We also obtain the local mixing for generalized BMS measures on $Γ\backslash G$ including Haar measures.
Exponential mixing via invariant foliations and relatively Anosov homeomorphisms
该研究构造了一类闭流形上的保体积同胚，实现了Hölder可观测量的相关性指数衰减，肯定了Dolgopyat和Pesin的相关猜想，还给出了无穷多个无对应微分同胚的此类流形实例。
Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups
本文为几何有限收敛群作用相关流空间上的有限BMS测度建立了计数、混合与等分布结果，并证明其适用于相对Anosov群相关的流空间。
A cohomological approach to Ruelle-Pollicott resonances and speed of mixing of Anosov diffeomorphisms
通过定义各向异性de Rham上同调并证明其与标准上同调同构，揭示了Anosov微分同胚的Ruelle-Pollicott共振与动力系统在de Rham上同调上诱导作用的特征值之间的深刻联系，并由此得到混合速度的上同调界。
We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…
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 article, we develop a geometric framework to study the notion of semi-minimality for the generic type of a smooth autonomous differential equation $(X,v)$, based on the study of rational factors of $(X,v)$ and of algebraic foliations on $X$, invariant under the Lie-derivative of the vector field $v$. We then illustrate the effectiveness of these methods by showing that certain autonomous algebraic differential equation of order three defined over the field of real numbers --- more precisely, those associated to mixing, compact, Anosov flows of dimension three --- are generically disint…
Anosov flows in dimension 3: an outside look
本文从外部视角概述三维 Anosov 流的主要结果与问题，重点阐述其与拓扑、辛几何及切触结构的联系，并介绍 Barbot-Fenley 叶空间理论。
Pseudo-Anosov autoquivalances arising from Symplectic topology and their hyperbolic actions on stability conditions
本文研究树形箭图对应的N-卡拉比-丘范畴中彭纳型自等价的渐近行为，提出计算拉伸因子的方法，并证明其在稳定性条件空间上诱导作用的平移长度正下界，且实例达到该下界，实现双曲作用。
Perturbation of the time-1 map of a generic volume-preserving $3$-dimensional Anosov flow
研究在C^s拓扑下接近体积保持3维阿诺索夫流时间-1映射的微分同胚，证明其动力学性质，包括混合性、唯一物理测度及无周期点的示例。
We define the prequantization of a symplectic Anosov diffeomorphism f:M-> M, which is a U(1) extension of the diffeomorphism f preserving an associated specific connection, and study the spectral properties of the associated transfer operator, called prequantum transfer operator. This is a model for the transfer operators associated to geodesic flows on negatively curved manifolds (or contact Anosov flows). We restrict the prequantum transfer operator to the N-th Fourier mode with respect to the U(1) action and investigate the spectral property in the limit N->infinity, regarding the transfer …
Exponential mixing of frame flows for three dimensional manifolds of quarter-pinched negative curvature
研究了三维四分针尖负曲率流形的框架流指数混合性，通过构造Young塔模型和加强Dolgopyat估计，扩展到Anosov流的torus扩展。
Mixing Rates for Analytic Observables: Toral Endomorphisms and Anosov Blaschke Deformations
该研究分析解析观测值的混合率，构造两类Anosov Blaschke形变，证明真正非线性系统存在双指数混合，且尖锐指数不连续，还为环面自同构及高阶作用建立双指数界。
Starting from any pseudo-Anosov map $φ$ on a surface of genus $g \geqslant 2$, we construct explicitly a family of Derived from pseudo-Anosov maps $f$ by adapting the construction of Smale's Derived from Anosov maps on the two-torus. This is done by perturbing $φ$ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure $μ$, supported by the non-trivial unique minimal component of the stable foliation of $f$, with respect to which $f$ is mixing. In the process, we construct a uniquely ergodic Ge…
Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces
本文证明了连通半单实代数群上Zariski稠密Anosov子群商空间中，单参数对角流关于Bowen--Margulis--Sullivan测度的局部混合，给出了带横截偏差的相关函数指数衰减一致界，并支撑了矩阵系数衰减的渐近公式应用。
We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M, is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in [6].
Standardly embedded train tracks and pseudo-Anosov maps with minimum expansion factor
本文证明了全穿孔伪Anosov映射的扩张因子下界 $λ(f)^{|χ(S)|} \ge μ^4 \approx 6.85408$，其证明基于对标准嵌入火车轨道及其权重空间上Thurston辛形式的研究。
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.
We show that the integrated Lyapunov exponents of $C^1$ volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all $C^1$ symplectic diffeomorphisms on…
Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing
本文通过构造非空间断域，证明无挠射影 Anosov 子群是具有唯一基本双曲集的局部齐次切触 Axiom A 流的单值群，并利用谱估计和亚纯延拓结果建立了指数混合等动力学性质。
Anosov groups: local mixing, counting, and equidistribution
本文研究了连通半单实代数群中 Anosov 子群的矩阵系数渐近行为，引入了 Burger-Robin 测度的高秩类比，并应用于仿射对称空间中离散轨道的双扇区计数。
We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the exponents in the center bundle vanish. This establishes in full a result announced by R. Mañé in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.
We investigate a class of area preserving non-uniformly hyperbolic maps of the two torus. First we establish some results on the regularity of the invariant foliations, then we use this knowledge to estimate the rate of mixing.
On a family of pseudo-Anosov-like maps on the infinite ladder surface
研究无限阶梯曲面\(\mathcal{L}\)上“拟伪阿诺索夫类”映射，通过\(S_g\)上彭纳型拟伪阿诺索夫映射在覆盖\(q_g\)下的提升得到，证明其拓扑传递、混合且支持零递归动力学，还给出具体例子。
Cyclotomic polynomials and homological criteria for mapping class types
研究闭可定向曲面\(S_g\)上由代数有限型映射类产生的整辛矩阵特征多项式，给出分圆情形下的刻画，即\(n\geq 3\)时，\(\varphi_n(x)\)由代数有限型映射类实现的条件，还得出相关准则及结论。
The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies …
On the variational principle for a class of skew product transformations
本文针对满足特定性质的唯一遍历同胚驱动的斜积变换，利用纤维指定性质证明两类变分原理，验证其具有Gundlach-Kifer指定性质，得到Hölder势的唯一平衡态并给出相关系统实例。
Emergence of quantum dynamics from chaos: The case of prequantum cat maps
本文提出了一种新的量子化理论，通过研究预量子转移算子的共振，揭示了猫映射中量子动力学与混沌之间的关系。
Construction of equilibrium states revisited
本文改进了Parmenter和Pollicott的抽象判据，将平衡态的几何构造推广到更广泛的微分同胚类，包括指数混合、Katok映射及几乎阿诺索夫情形，并应用于齐次空间上的部分双曲仿射系统。
We study Poincaré recurrence for flows and observations of flows. For Anosov flow, we prove that the recurrence rates are linked to the local dimension of the invariant measure. More generally, we give for the recurrence rates for the observation an upper bound depending on the push-forward measure. When the flow is metrically isomorphic to a suspension flow for which the dynamic on the base is rapidly mixing, we prove the existence of a lower bound for the recurrence rates for the observation. We apply these results to the geodesic flow and we compute the recurrence rates for a particular obs…
Rigidity of equilibrium states and unique quasi-ergodicity for horocyclic foliations
本文证明拓扑混合度量Anosov流的平衡态由稳定/不稳定叶上的分解唯一决定，进而推出水平叶理流的唯一拟遍历性，并给出Babillot-Ledrappier结果的另一证明。
We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the opposite, it is …
We study diffusion and mixing in different linear fluid dynamics models, mainly related to incompressible flows. In this setting, mixing is a purely advective effect which causes a transfer of energy to high frequencies. When diffusion is present, mixing enhances the dissipative forces. This phenomenon is referred to as enhanced dissipation, namely the identification of a time-scale faster than the purely diffusive one. We establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales. The proofs are based on a …
Examples beyond Bounded Mean Motion for Quantitative Rigidity on the Two-Torus
该研究为双环面上定量刚性的Sarnak猜想相关论文，构造了不满足有界平均运动的半无理及全无理微分同胚例子，证明其数量为连续统多个。
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