Israel Or Weinstein
We prove that for any holomorphic map, and any bounded orbit which does not accumulate to a singular set or to an attracting cycle, its lower Lyapunov exponent is non-negative. The same result holds for unbounded orbits, for maps with a bounded singular set. Furthermore, the orbit may accumulate to infinity or to a singular set, as long as it is slow enough.
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Pointwise Hölder continuity of Lyapunov exponents of random matrix products
该研究证明紧支撑随机矩阵乘积的简单第一李雅普诺夫指数关于Wasserstein-Hausdorff距离逐点赫尔德连续，且前k个简单指数可通过外幂推导得同样性质。
For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $μ$ is an arbitrary $f$-invariant measure with positive Lyapunov exponents on the boundary of $A$, then $μ$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessability of every "good" $q$ i.e. such $q$ for which "small neighbourhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem …
We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of CP(k) and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic theory and on…
Pointwise modulus of continuity of the Lyapunov exponent and integrated density of states for analytic multi-frequency quasiperiodic $M(2, \mathbb{C})$ cocycles
本文针对解析多频准周期余环，建立了 Lyapunov 指数在余环和频率上的逐点模量连续性，并由此得到态密度积分在能量、势和频率上的正则性。
We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.
Let \Fc be a holomorphic foliation by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: (1) the singular points of \Fc are all hyperbolic; (2) \Fc is Brody hyperbolic. Then we establish cohomological formulas for the Lyapunov exponent and the Poincaré mass of an extremal positive \ddc-closed current tangent to \Fc. If, moreover, there is no nonzero positive closed current tangent to \Fc, then we show that the Lyapunov exponent χ(\Fc) of \Fc, which is, by definition, the Lyapunov exponent of the unique normalized positive \ddc-closed cur…
A superposition approach for the ISS Lyapunov-Krasovskii theorem with pointwise dissipation
本文提出一种叠加方法，通过李雅普诺夫-克拉索夫斯基函数的点wise耗散性来保证输入到状态稳定性，同时利用该函数确保全局统一稳定性。
We show that typical (in the sense of Bonatti-Viana) Hölder and fiber-bunched $GL_d(\mathbb{R})$-valued cocycles over a subshift of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.
Generalizing results of Deroin-Dujardin, we introduce the notion of proximal stability for a holomorphic family $ (ρ_λ) $ of representations $ Γ\to \text{SL}(d, \mathbb{C}) $, where $ Γ$ is a finitely generated group, and show that it is equivalent to the pluriharmonicity of Lyapunov exponents of the family (defined using random walks).
We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k exponents is greater or equal than the sum of the degree of any rank k holomorphic subbundle of the flat bundle and the asymptotic degree of its equivariant developing map. We also show that this inequality is an equality if the base curve is compact. We moreover relate the asymptotic degree to the dynamical degree defined by Daniel and Deroin. We then use the p…
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。
Lyapunov Exponent and Stochastic Stability for Infinitely Renormalizable Lorenz Maps
本文研究无限可重整压缩 Lorenz 映射，证明在有界几何或先验界条件下其临界值满足慢回归，并据此得到零逐点 Lyapunov 指数与随机稳定性。
Regularity of Lyapunov exponents at one-point Lyapunov spectra: the semisimple case
本文研究了Lyapunov指数作为紧支撑概率测度空间上的函数的正则性，证明了在半单概率测度具有单点Lyapunov谱时，Lyapunov指数在Wasserstein距离下点wise log-Hölder连续。
Analyticity of the Lyapunov exponents of random products of matrices
本文研究随机矩阵乘积中李亚普诺夫指数的解析性，结合马尔可夫算子谱性质与巴纳赫空间全纯函数理论，扩展了有限到无限紧致符号空间上的结果。
Log-Höder continuity at zero Lyapunov gap for finite state Markov $GL(2)$-cocycles
该研究证明有限状态马尔可夫GL(2)上同调在零李雅普诺夫间隙处的极值李雅普诺夫指数满足对数-赫尔德连续性，提出非分裂三角情形的马尔可夫永续性估计方法，共形情形下对数模指数可优化为1。
We study holomorphic families of polynomial-like maps depending on a parameter s. We prove that the partial sums of largest Lyapunov exponents are plurisubharmonic functions of s. We also study their continuity and introduce the bifurcation locus as the support of bifurcation currents.
We consider the case of hyperbolic basic sets $Λ$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $μ_ϕ$ associated to a class of Hölder potentials $ϕ$ on $Λ$, and find the measures $μ_ϕ$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $δ_{μ_ϕ}$ of $μ_ϕ$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $μ_ϕ$ in the case when the preimage counting function is constant on $Λ$. For terminal/minimal saddle sets we prove that an invariant…
If E is a flat bundle of rank r over a Kähler manifold X, we define the Lyapunov spectrum of E: a set of r numbers controlling the growth of flat sections of E, along Brownian trajectories. We show how to compute these numbers, by using harmonic measures on the foliated space P(E). Then, in the case where X is compact, we prove a general inequality relating the Lyapunov exponents and the degrees of holomorphic subbundles of E and we discuss the equality case.
On the Lyapunov spectrum of the twisted cocycle for substitutions
本文基于Rajabzadeh和Safaee发现的扭曲上同调不变截面，证明代换动力系统存在零李雅普诺夫指数，简化双符号代换奇异谱证明并得到最大指数正性结果，附录关联谱测度局部维数与李雅普诺夫指数。
Hölder regularity of the top Lyapunov exponent for random matrix products with low-dimensional equator
该研究针对任意维度d≥2的随机可逆矩阵乘积，证明了当最大李雅普诺夫指数简单且赤道维度≤1时，其在紧支撑概率测度空间中关于Wasserstein-Hausdorff距离逐点赫尔德连续，完善了相关正则性理论。
Continuity of Lyapunov exponents for stochastic differential equations
本文首次系统研究了随机微分方程Lyapunov指数的连续性，在系数几乎一致收敛、逐点收敛及严格单调性等不同条件下，分别建立了顶部或全部Lyapunov指数的连续性、Lipschitz连续性与Hölder连续性。
Bifurcation loci in the moduli space of degree $d$ rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period $n$ and multiplier 0 or $e^{iθ}$. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree $d=2$, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin of fixed period exists.
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in di…
Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps
本研究证明$\u2119^k(\u2102)$自同态稳定族中满足熵条件的不变测度可全纯延拓，推广了已有结果，并将类似结论拓展至大拓扑度类多项式映射族，给出Lyapunov指数正性的充分条件。
Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $α_i$ and $β_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge str…
Integral representation of Lyapunov exponents
本文提出一个抽象算子变分原理，用于描述由马尔可夫算子驱动的亚加性过程的渐近增长率，扩展了经典投影公式，并给出了新的渐近表示。
We study the holonomy cocycle H of a holomorphic foliation \Fc by Riemann surfaces defined on a compact complex projective surface X satisfying the following two conditions: 1) its singularities E are all hyperbolic; 2) there is no holomorphic non-constant map \C\to X such that out of E the image of \C is locally contained in a leaf. Let T be a harmonic current tangent to \Fc which does not give mass to any invariant analytic curve. Using the leafwise Poincaré metric, we show that H is integrable with respect to T. Consequently, we infer the existence of the Lyapunov exponent function of T.
On invariant measures of "satellite" infinitely renormalizable quadratic polynomials
本文证明具有无限卫星重整化序列的二次多项式在 J_∞ 上的所有不变测度均支撑于后临界集且 Lyapunov 指数为零，从而部分回答了 Weixiao Shen 提出的问题。
On the parameter space of fibered hyperbolic polynomials
本文将拟共形手术应用于准周期受迫全纯映射，通过修改吸引不变曲线的纤维化乘子，证明了具有丢番图频率的QPF二次映射族的纤维化乘子映射是复Banach流形上的全纯浸没。
Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, …
We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on Markovian rules. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the Lyapunov exponent is uniformly negative for every initial value and almost every random orbit. Moreover, we consider families of random holomorphic dynamical systems and show that the set of mean stable systems has full measure under certain conditions. The latter is a new result even for i.i.d. random dynamical systems.
For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a Hénon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any Hénon map is accumulated by the bifurcation locus in the space of quadratic holomorphic end…
Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights
本文证明Hecke特征值平方与Piltz除数函数可作为逐点遍历定理的良好权重，部分解决Cuny和Weber的问题，并推广至其他算术函数。
Convex computation of maximal Lyapunov exponents
本文提出通过带平方和约束的多项式优化凸松弛方法，从上方逼近ODE动力系统在指定集合上的最大李雅普诺夫指数，并在Lorenz和Hénon-Heiles混沌系统上验证了上界可精确到至少五位数字。
The symmetric group S_n acts as a reflection group on CP^{n-2} (for $n\geq 3$) . Associated with each of the $\binom{n}{2}$ transpositions in S_n is an involution on CP^{n-2} that pointwise fixes a hyperplane--the mirrors of the action. For each such action, there is a unique S_n-symmetric holomorphic map of degree n+1 whose critical set is precisely the collection of hyperplanes. Since the map preserves each reflecting hyperplane, the members of this family are critically-finite in a very strong sense. Considerations of symmetry and critical-finiteness produce global dynamical results: each m…
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponen…
A Mañé-Manning formula for expanding measures for endomorphisms of $\mathbb P^k$
本文为 $\mathbb P^k$ 上具有严格正 Lyapunov 指数的遍历测度引入体积维数，证明推广的 Mañé-Manning 公式，并刻画自然压力函数的第一个零点，将 1 维结果推广到任意维数。
This report discusses recent results as well as new perspectives in the ergodic theory for Riemann surface laminations, with an emphasis on singular holomorphic foliations by curves. The central notions of these developments are leafwise Poincaré metric, directed positive harmonic currents, multiplicative cocycles and Lyapunov exponents. We deal with various ergodic theorems for such laminations: Random and Operator Ergodic Theorems, (Geometric) Birkhoff Ergodic Theorems, Oseledec Multiplicative Ergodic Theorem and Unique Ergodicity Theorems. Applications of these theorems are also given. In p…
Limit cycles in piecewise smooth systems with circular switching manifold
研究分段光滑系统中极限环的性质，利用莫比乌斯变换建立圆状与直线状不连续性的等价性，通过二阶平均和李雅普诺夫量得出极限环数量的下界，并证明分段反全纯系统中极限环的上界，同时构造了显式代数极限环。
A Lyapunov exponent attached to modular functions
本研究针对满足特定非负条件的SL(2,Z)弱全纯模函数，构造推广已知形式的GL(2,Z)不变李雅普诺夫指数映射，证明其值域性质、在马尔可夫无理数上的递增凸性，及纯周期二次无理数处的值与模曲面周期积分的关系，关联Kaneko关于j函数的猜想。
We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values $z$ in the Riemann sphere, for almost every sequence of maps $γ=(γ_{1},γ_{2},\ldots )$, the Lyapunov exponent of $γ$ at $z$ is negative. Also, we show that for a generic system, for every initial value $z$ in the Riemann sphere, the or…
The dimension of harmonic currents on foliated complex surfaces
本文研究代数复曲面上奇异全纯叶状结构的调和流维数，给出用 Furstenberg 熵和 Lyapunov 指数表示的横向豪斯多夫维数公式，并将 Brunella 不等式推广到任意全纯叶状结构，同时确定了 Jouanolou 叶状结构的维数为 1/4。
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