Jérôme Buzzi, Sylvain Crovisier, Omri Sarig
Recently, Burguet proved a strong form of Viana's conjecture on physical measures, in the special case of $C^\infty$ surface diffeomorphisms. We give another proof, based on our analysis of entropy and Lyapunov exponents in [BCS].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Inverse SRB measures for endomorphisms on surfaces
本文研究了二维流形上内映射的逆SRB测度，通过构造最大化折叠熵的测度，得到了具有至少两个逆SRB测度的拓扑混合映射的例子，并给出了唯一性的判据。
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.
A $C^\infty$ surface diffeomorphism admits a SRB measure if and only if the set \left \{x, \limsup_n \frac{1}{ n} \log \|d_xf^n \|> 0\right\} has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for $C^r$ surface diffeomorphisms with $+\infty>r>1$.
We study the existence of SRB measures of C 2 diffeomorphisms for attractors whose bundles admit Holder continuous invariant (non-dominated) splittings. We prove the existence when one subbundle has the non-uniform expanding property on a set with positive Lebesgue measure and the other subbundle admits non-positive Lyapunov exponents on a total probability set.
In the present paper we contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures…
We consider a diffeomorphism f of a compact manifold M which is Almost Axiom A, i.e. f is hyperbolic in a neighborhood of some compact f-invariant set, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure such that for every point in this set the stable and unstable leaves are "long enough", then f admits a probability SRB measure.
We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+α}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original method for constructing first return Young towers, proving that every hyperbolic measure, and in particular every SRB measure, can be lifted to such a tower. This method relies on a new general result on hyperbolic branches and shadowing for pseudo-orbits in nonuniformly hyperbolic sets which is of independent interest.
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…
For a C^{1+α} diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μcan be approximated by a sequence of hyperbolic sets \{Λ_n\}_{n\geq1}. In this paper, we prove the Hausdorff dimension for Λ_n on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of μcan be approximated by the Hausdorff dimension of Λ_n. To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of -ψ^{s}(\cdot,f^n) and…
SRB Measures for $C^{1+\mathrm{Dini}}$ Diffeomorphisms
研究\(C^{1 + \mathrm{Dini}}\)微分同胚，证明其SRB测度刻画，当支配丛有正Lyapunov指数时条件测度绝对连续性与部分Pesin熵公式等价，得到部分双曲大多扩张吸引子的SRB测度，还指出\(C^1\)范畴不成立。
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。
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。
In this paper we give an upper bound for the number of SRB measures of saddle type of local diffeomorphisms of boundaryless manifolds in terms of maximal cardinality of set of periodic points without any homoclinic relation.
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.
Measures of maximal entropy that are SRB
本文研究$T^3$上DA-微分同胚的最大熵测度的SRB性质，给出其为SRB测度的充要条件（正Lyapunov指数和在支撑集周期轨道上与线性Anosov映射一致），并指出该刚性结果弱于Katok猜想且附示例。
We construct SRB measures for endomorphisms satisfying conditions far weaker than the non-uniformly expansion. As a consequence, the definition of non-uniformly expanding map can be weakened. We also prove the existence of an absolutely continuous invariant measure for local diffeomorphisms, only assuming the existence of hyperbolic times for Lebesgue almost all point of the manifold.
Exponentially mixing SRB measures are Bernoulli
本文证明了紧流形上C^{1+α}微分同胚所保持的SRB测度，若具有指数混合性或由指数体积极限得到，则该测度必为伯努利测度。
Let $M$ be a Riemannian, boundaryless, and compact manifold, with $\dim M\geq 2$ and let $f$ be a $C^{1+}$ diffeomorphism. We show that there is a hyperbolic SRB measure if and only if there exists an unstable leaf with a subset of positive leaf volume of hyperbolic points which return to some Pesin set with positive frequency. This answers a question of Pesin.
Optimal linear response for Anosov diffeomorphisms
本文研究Anosov微分同胚在微小扰动下的最优线性响应，通过转移算子在各向异性Banach空间中的响应理论，提出优化SRB期望的方法，并给出最优扰动的显式表达式及收敛证明。
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…
We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an "effective hyperbolicity" condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.
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。
In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one $SRB$ measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including …
We prove that a partially hyperbolic attractor for a $C^1$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $C^2$, and the center bundle admits the sectional expanding condition w.r.t. any Gibbs $u$-state, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.
We give a geometric proof, offering a new and quite different perspective on an earlier result of Ledrappier and Young on random transformations. We show that under mild conditions, sample measures of random diffeomorphisms are SRB measures. As sample measures are the limits of forward images of stationary measures, they can be thought of as the analog of physical measures for deterministic systems. Our results thus show the equivalence of physical and SRB measures in the random setting, a hoped-for scenario that is not always true for deterministic maps.
We prove the abundance of Sinai-Ruelle-Bowen measures for diffeomorphisms away from ones with a homoclinic tangency. This is motivated by conjectures of Palis on the existence of physical (Sinai-Ruelle-Bowen) measures for global dynamics. The main novelty in this paper is that we have to deeply study Gibbs $cu$-states in different levels. Note that we have to use random perturbations to give some upper bound of the level of Gibbs $cu$-states.
This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $ρ$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $ρ$. We propose that generically, if $ρ$ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of $ρ(ϕ)$ with respect to $f$ along $X$ is convergent: $$ Ψ(z)=\sum_{n=0}^\infty z^n\intρ(dx)\,X(x)\cdot\nabla_x(ϕ\circ f^n)\qq…
We consider partially hyperbolic \( C^{1+} \) diffeomorphisms of compact Riemannian manifolds of arbitrary dimension which admit a partially hyperbolic tangent bundle decomposition \( E^s\oplus E^{cu} \). Assuming the existence of a set of positive Lebesgue measure on which \( f \) satisfies a weak nonuniform expansivity assumption in the centre~unstable direction, we prove that there exists at most a finite number of transitive attractors each of which supports an SRB measure. As part of our argument, we prove that each attractor admits a Gibbs-Markov-Young geometric structure with integrable…
We consider $C^2$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volum…
Entropy formula for surface diffeomorphisms
本文推导了表面微分同胚的拓扑熵公式，并通过曲线体积增长特性进行刻画，展示了其在动力系统中的应用。
Emergence for diffeomorphisms with nonzero Lyapunov exponents
研究保持双曲测度的 $C^{1+α}$ 微分同胚的高点态涌现性点集，基于不稳定Hausdorff维数给出其维数下界，并证明SRB测度情形下该点集具有满Hausdorff维数。
In this paper, we study the differentiability of SRB measures for partially hyperbolic systems. We show that for any $s \geq 1$, for any integer $\ell \geq 2$, any sufficiently large $r$, any $φ\in C^{r}(\T, \R)$ such that the map $f : \T^2 \to \T^2, f(x,y) = (\ell x, y + φ(x))$ is $C^r-$stably ergodic, there exists an open neighbourhood of $f$ in $C^r(\T^2,\T^2)$ such that any map in this neighbourhood has a unique SRB measure with $C^{s-1}$ density, which depends on the dynamics in a $C^s$ fashion. We also construct a $C^{\infty}$ mostly contracting partially hyperbolic diffeomorphism $f: \T…
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.
We study the entropy and Lyapunov exponents of invariant measures $μ$ for smooth surface diffeomorphisms $f$, as functions of $(f,μ)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion f…
We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under $C^1$ perturbations of the map. These results are an application of general machinery developed by the first and last named authors.
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…
We present conditions on families of diffeomorphisms that guarantee statistical stability and SRB entropy continuity. They rely on the existence of horseshoe-like sets with infinitely many branches and variable return times. As an application we consider the family of Henon maps within the set of Benedicks-Carleson parameters.
A generalized push-forward construction of Sinai-Ruelle-Bowen measures
本文针对正体积集合上有效双曲的微分同胚，提出SRB测度构造中参考测度的广义选取方法，其结果还适用于均匀部分双曲或具主导分裂的微分同胚。
Open sets of partially hyperbolic skew products having a unique SRB measure
本文获得了耗散部分双曲斜积的$C^2$-开集，其具有唯一满支撑和满盆地的SRB测度，核心是通过改编测度刚性结果对邻域内的$u$-Gibbs测度进行分类并排除部分可能性。
We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov diffeomorphisms --- a conservative family and a dissipative one. For diffeomorphisms in these families we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in $\mathbb T^3$. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifo…
Transfer Operators and SRB Measures for Axiom A Diffeomorphisms: Spectral Gap, Structural Stability, and the Gibbs Equivalence Theorem
本文研究Axiom A微分同胚的转移算子和SRB测度，通过马尔可夫分区编码将符号谱结果推广到光滑动力学，提出四个主要定理，包括结构稳定性、转移算子的拟紧致性、SRB测度的构造以及佩辛熵公式，最终通过四个主要定理和导入结果证明吉布斯等价定理。
Finiteness of physical measures for diffeomorphisms with multi 1-D centers
该研究针对具有多1维中心的部分双曲$C^2$微分同胚，在Gibbs $u$-态双曲性的条件下，证明了其遍历物理（SRB）测度具有有限性，推进了部分双曲系统的测度理论研究。
On positive Lyapunov exponents and SRB measures for partially hyperbolic systems
研究紧致黎曼流形上部分双曲系统，通过可观测量测度计算沿$E^{cu}$的最小李雅普诺夫指数，证明特定条件下$f$沿$E^{cu}$非一致扩张，得出满足条件的$C^{1 + \alpha}$微分同胚有SRB测度的结论。
In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin-Sinai's notion for partially hyperbolic attractors of $C^{1+α}$ diffeomorphisms. In particular, we prove that for partially hyperbolic $C^{1+α}$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point: (1) the center Lyapunov exponent is well defined, but (…
Decay of Correlations for some Non-Uniformly Hyperbolic Attractors
针对光滑实现问题，研究非一致双曲吸引子动力系统的关联衰减，通过对一致双曲微分同胚施加减速程序构造系统，证明其关联存在多项式上下界，结果适用于Smale-Williams螺线及其小扰动。
Continuity properties of partial entropy
本文研究了$C^{1+α}$微分同胚中偏熵的上半连续性，证明了当Lyapunov指数之和连续时该性质成立，扩展了Newhouse关于光滑微分同胚的结果。
Smooth rigidity for 3-dimensional dissipative Anosov flows
本文证明三维耗散Anosov流在连续共轭下，除例外情形外，共轭或光滑或交换正负SRB测度，并由此建立局部刚性、改进经典定理并揭示Teichmüller空间的分层结构。
We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003), 25--62, where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak* topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence…
Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$
本文证明了三维环面上保守 Anosov 微分同胚的 $C^1$ 邻域内，任意 $C^2$ 微分同胚要么强稳定与强不稳定丛联合可积，要么其所有全支撑 $u$-Gibbs 测度均为 SRB 测度。
Given a surface $M$ and a Borel probability measure $ν$ on the group of $C^2$-diffeomorphisms of $M$, we study $ν$-stationary probability measures on $M$. Assuming the positivity of a certain entropy, the following dichotomy is proved: either the stable distributions for the random dynamics is non-random, or the measure is SRB. In the case that $ν$-a.e. diffeomorphism preserves a common smooth measure $m$, we show that for any positive-entropy stationary measure $μ$, either there exists a $ν$-almost surely invariant $μ$-measurable line field (corresponding do the stable distributions for almos…
We consider a smooth Anosov diffeomorphism with a smooth dynamical foliation. We show upper bounds on the essential spectral radius of its transfer operator acting on anisotropic Sobolev spaces. (Such bounds are related to the essential decorrelation rate for the SRB measure.) We compare our results to the estimates of Kitaev on the domain of holomorphy of dynamical Fredholm determinants for differentiable dynamics.
Let f be a self-map of a compact manifold M, admitting an global SRB measure μ. For a continuous test function ϕon M and a constant α>0, consider the set of the initial points for which the Birkhoff time averages of the function ϕdiffer from its μ--space average by at least α. As the measure μis an SRB one, the intersection of this set with the basin of attraction of μshould have zero Lebesgue measure. The \emph{special ergodic theorem}, whenever it holds, claims that, moreover, this intersection has the Hausdorff dimension less than the dimension of M. We prove that for Lipschitz maps, the sp…
Given a surface $M$ and a Borel probability measure $ν$ on the group of $C^2$-diffeomorphisms of $M$, we study $ν$-stationary probability measures on $M$. We prove for hyperbolic stationary measures the following trichotomy: either the stable distributions are non-random, the measure is SRB, or the measure is supported on a finite set and is hence almost-surely invariant. In the proof of the above results, we study skew products with surface fibers over a measure preserving transformations equipped with a decreasing sub-$σ$-algebra $\hat {\mathcal F}$ and derive a related result. A number of a…
Multifractal Analysis, Livšic Rigidity, and Fluctuation Theorems for Axiom A Diffeomorphisms: The Pesin Formula and the Gallavotti-Cohen Symmetry
本文研究Axiom A微分同胚的多标度分析、Livšic定理及波动定理，推导了Pesin熵公式和Gallavotti-Cohen对称性，并探讨了压力的Legendre变换与Hausdorff维数的关系。
We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the map $R(t)$ sending $t$ to the average of $G$ with respect to the SRB measure of $f_t$ is $α$-Hölder continuous for all $α<r- |log \mathcal J|/(p|log ν_s|$) where $\mathcal J\le 1$ is the strongest volume contraction and $ν_s<1$ is the weakest contraction. This applies to $θ(x)=h(x)Θ(g(x)-a)$ (for all $α<1- |log \mathcal J|/|log ν_s|$) for $h$ and $g$ smooth and $…
Physical measures for mostly sectional expanding flows
针对C2向量场的部分双曲吸引集，证明其支撑遍历物理/SRB测度的充要条件为捕获区域在正勒贝格测度子集上存在非一致截面扩张，最多支撑有限个此类测度且为中心不稳定方向吉布斯态，推广微分同胚结果至连续时间系统并给出应用实例。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。