Romain Dujardin
We prove two continuity theorems for the Lyapunov exponents of the maximal entropy measure of polynomial automorphisms of $\mathbb{C}^2$. The first continuity result holds for any family of polynomial automorphisms of constant dynamical degree. The second result is the continuity of the upper exponent for families degenerating to a 1-dimensional map.
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We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.
In this paper we establish uniform large deviations estimates of exponential type and Hölder continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/Γ$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$ whenever $f$ is $C^{1+}$ and has the same volume Lyapunov spectrum as $L$. For $L$ with simple spectrum, we show that local Lyapunov spectrum rigidity is equivalent to $L$ satisfying both an irreducibility condition and an ordering condition on its Lyapunov exponents.
Entropy Continuity of Lyapunov Exponents for Non-flat 1-dimensional Maps
该研究证明了非平坦一维映射中李雅普诺夫指数的熵连续性，并展示了其在熵上的统一可积性。
The Lyapunov exponents of locally constant GL(2;C)-cocycles over Bernoulli shifts depend continuously on the cocycle and on the invariant probability. The Oseledets decomposition also depends continuously on the cocycle, in measure.
We provide conditions which imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb{R})$. The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.
We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the Hölder exponent is independent of the frequency and the coupling constant.
We prove that the Bochi-Mañé theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any vo…
We prove a conjecture of Viana which states that Lyapunov exponents vary continuously when restricted to $GL(2,\mathbb{R})$-valued cocycles over a subshift of finite type which admit invariant holonomies that depend continuously on the cocycle.
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。
We study the regularity of the conjugacy between an Anosov automorphism $L$ of a torus and its small perturbation. We assume that $L$ has no more than two eigenvalues of the same modulus and that $L^4$ is irreducible over $\mathbb Q$. We consider a volume-preserving $C^1$-small perturbation $f$ of $L$. We show that if Lyapunov exponents of $f$ with respect to the volume are the same as Lyapunov exponents of $L$, then $f$ is $C^{1+\text{Hölder}}$ conjugate to $L$. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.
We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.
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…
This paper is concerned with the study of linear cocycles over uniformly ergodic Markov shifts on a compact space of symbols. We establish the joint Hölder continuity of the maximal Lyapunov exponent as a function of the cocycle and the transition kernel in the vicinity of any irreducible cocycle with simple maximal Lyapunov exponent. Our approach, via Furstenberg's formula, shows the Hölder continuous dependence on the data of the stationary measure of the projective cocycle and in particular provides a more computable Hölder exponent.
Continuity of Lyapunov exponents for C^r one-dimensional maps
本文研究区间或圆周 C^r 一维映射，证明大熵条件下 Lyapunov 指数的熵连续性，并推出熵的上半连续性与几何势的一致可积性。
Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Di…
We collect some applications of the variational formula established by Schröder (1988) and Rueß(2013) for the quenched Lyapunov exponent of Brownian motion in stationary and ergodic nonnegative potential. We show for example that the Lyapunov exponent for nondeterministic potential is strictly lower than the Lyapunov exponent for the averaged potential. The behaviour of the Lyapunov exponent under independent perturbations of the underlying potential is examined. And with the help of counterexamples we are able to give a detailed picture of the continuity properties of the Lyapunov exponent.
Continuity of the Lyapunov exponents of random matrix products
本文证明随机矩阵乘积的Lyapunov指数在矩阵系数、概率权重及分布（在弱*和Hausdorff拓扑下）连续变化，推广了已有结果。
We prove that a locally constant $SL_{2}(\mathbb{R})$-valued cocycle over the shift generated by an irreducible collection of matrices is a continuity point for Lyapunov exponents in the $α$-Hölder topology for every $α> 0$. This gives negative answers to conjectures of Viana and the author; we pose a new conjecture to replace these conjectures. We show that an analogous continuity result also holds for $GL_{2}(\mathbb{R})$-valued cocycles that admit canonical holonomies.
Hölder continuity of Lyapunov exponents for non-invertible and non-compact random cocycles
本文在有限指数矩、谱间隙和拟不可约性三个条件下，证明了由独立同分布过程驱动的随机线性余循环的最大李雅普诺夫指数关于Wasserstein距离具有Hölder连续性，并将其应用于无界位势的Schrödinger余循环。
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。
We study regularity properties of the Lyapunov exponent L of quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove that L is jointly continuous, in frequency and energy, at every irrational frequency.
We construct discontinuous point of the Lyapunov exponent of quasiperiodic Schrödinger cocycles in the Gevrey space $G^{s}$ with $s>2$. In contrast, the Lyapunov exponent has been proved to be continuous in the Gevrey space $G^{s}$ with $s<2$ \cite{klein,cgyz}. This shows that $G^2$ is the transition space for the continuity of the Lyapunov exponent.
We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $C^{k}$ quasi-periodic $SL(2,\mathbb{R})$ cocycle is $1/2$-Hölder continuous in the almost reducible regime, if $k$ is large enough. As a consequence, we show that if the frequency is Diophantine, $k$ is large enough, and the potential is $C^k$ small, then the integrated density of states of the corresponding quasi-periodic Schrödinger operator is $1/2$-Hölder continuous.
We compute entropies in a large family of K3 surface automorphisms in (P^1)^3. In keeping with a result by Xie, we find that the entropies vary in a lower semi-continuous manner as the Picard ranks of the K3 surfaces vary.
We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008, Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp $\frac12$-Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp $\frac12$-Hölder continuous below the lowest energy of the spectrum, in the large cou…
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.
Let $(X_{A},σ_{A})$ be a shift of finite type and $\text{Aut}(σ_{A})$ its corresponding automorphism group. Associated to $ϕ\in \text{Aut}(σ_{A})$ are certain Lyapunov exponents $α^{-}(ϕ), α^{+}(ϕ)$ which describe asymptotic behavior of the sequence of coding ranges of $ϕ^{n}$. We give lower bounds on $α^{-}(ϕ), α^{+}(ϕ)$ in terms of the spectral radius of the corresponding action of $ϕ$ on the dimension group associated to $(X_{A},σ_{A})$. We also give lower bounds on the topological entropy $h_{top}(ϕ)$ in terms of a distinguished part of the spectrum of the action of $ϕ$ on the dimension gr…
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 show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schrödinger cocycles are uniformly positive and weak Hölder continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak Hölder continous. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general $\mathrm{SL}(2,\mathbb R)$ cocycles, which in turn can be applied to get uniform positivit…
We study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show the continuity of the entropy spectrum at the boundary of Lyapunov spectrum in the sense that $h_{top}(E(α_{t}))\ \rightarrow h_{top}(E(β(\mathcal{A}))$ for generic cocycles, where $E(α)=\{x\in X: \lim_{n\rightarrow \infty}\frac{1}{n}\log \|\mathcal{A}^{n}(x)\|=α\}$. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles over mixing subshifts of finite type. Moreover, we prove the restricted variational principle…
Abstract Continuity Theorem for the Lyapunov Exponents of linear cocycles
本文建立线性余循环 Lyapunov 指数关于基测度的 Hölder 连续性，利用一致 LDT 估计、Urysohn 型引理和 Wasserstein 距离性质给出测度版 ACT，并与原 ACT 结合解决测度与纤维的联合连续性。
Lyapunov stability of the equilibrium of the non-local continuity equation
本文针对非局部连续性方程驱动的概率测度空间动力系统，通过分析非光滑 Lyapunov 函数推导平衡点稳定性的充分条件，并将线性动力学稳定性简化为切空间二次型研究，最后以梯度流和耦合数学摆系统为例验证。
We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses. Applications are one-parameter families of analytic Jacobi operators, such as extended Harper's model describing crystals subject to external magnetic fields.
The precise regularity of the Lyapunov exponent for $C^2$ Cos-type quasiperiodic Schrödinger cocycles with large couplings
研究大耦合下$C^2$ cos型拟周期薛定谔余环的Lyapunov指数，证明其绝对连续性和$\frac{1}{2}$-Hölder连续性，并刻画谱中局部正则性的精确范围。
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。
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。
Analiticity of the Lyapunov exponents of perturbed toral automorphisms
本文研究了环面上解析 Anosov 微分同胚在解析扰动下的动力系统，证明了不变子空间的分裂可通过部分共轭扩展至扰动系统，并得出非退化 Lyapunov 指数是扰动的解析函数。
Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting
本文研究具有控制分裂的紧流形$C^r(r>1)$微分同胚，在两类Lyapunov指数符号分离子丛的情形下，证明了度量熵映射的上半连续性，丰富了动力系统熵理论。
An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We prove large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we establish continuity properties of the Lyapu…
Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…
Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$, is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on $\mathbb{T}^3$.
In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus $\mathbb{T}^d, d \geq 3,$ $C^1-$close to a linear automorphism diagonalizable over $\mathbb{R}$ and such that its characteristic polynomial is irreducible over $\mathbb{Q}.$
We devise an abstract, modular scheme to prove continuity of the Lyapunov exponents for a general class of linear cocycles. The main assumption is the availability of appropriate large deviation type (LDT) estimates which are uniform in the data. We provide a modulus of continuity that depends explicitly on the sharpness of the LDT estimate. Our method uses an inductive procedure based on the deterministic, general Avalanche Principle in [4]. The main advantage of this approach, besides the fact that it provides quantitative estimates, is its versatility, as it applies to quasi-periodic cocycl…
We consider the quasi-periodic Jacobi operator $H_{x,ω}$ in $l^2(\mathbb{Z})$ $(H_{x,ω}ϕ)(n) = -b(x+(n+1)ω)ϕ(n+1) - b(x+nω)ϕ(n-1) + a(x+nω)ϕ(n) = Eϕ(n),\ n\in\mathbb{Z},$ where $a(x),\ b(x)$ are analytic function on $\mathbb{T}$, $b$ is not identically zero, and $ω$ obeys some strong Diophantine condition. We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $ρ_0=ρ_0(a,b,ω,E_0)$, $β=β(a,b,ω)$ such that $|L(E)-L(E')|<|E-E'|^β$ for any $E,E'\in (E_0-ρ_0,E_0+ρ_0)$. If $L(E)>0$ for all $E$ in …
In this paper we prove the continuity of all Lyapunov exponents, as well as the continuity of the Oseledets decomposition, for a class of irreducible cocycles over strongly mixing Markov shifts. Moreover, gaps in the Lyapunov spectrum lead to a Hölder modulus of continuity for these quantities. This result is an application of the abstract continuity theorems obtained in [4], and generalizes a theorem of E. Le Page on the Hölder continuity of the maximal LE for one-parameter families of strongly irreducible and contracting cocycles over a Bernoulli shift. This is a draft of a chapter in our fo…
In this paper we continue our study of polynomial diffeomorphisms of C^2. Let us recall that there is an invariant measure $μ$, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C. In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to $μ$ analogous to the Brolin-Manning formula polynomial maps of C. Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these critical points have a very natural …
Obstructions to the regularity of the Lyapunov exponents for non-compact random Schrödinger cocycles
该研究针对非紧支撑的随机薛定谔上循环，揭示其李雅普诺夫指数正则性的关键障碍，明确赫尔德连续性需满足指数矩条件。
We show that the ergodicity of an aperiodic automorphism of a Lebesgue space is equivalent to the continuity of a certain map on a metric Boolean algebra. A related characterization is also presented for periodic and totally ergodic transformations
We give a new proof of E. Le Page's theorem on the Holder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first Lyapunov exponent, as well as the stationary measure, for such random cocycles.
We provide an example of a Schrödinger cocycle over a mixing Markov shift for which the integrated density of states has a very weak modulus of continuity, close to the log-Hölder lower bound established by W. Craig and B. Simon. This model is based upon a classical example due to Y. Kifer of a random Bernoulli cocycle with zero Lyapunov exponents which is not strongly irreducible. It follows that the Lyapunov exponent of a Bernoulli cocycle near this Kifer example cannot be Hölder or weak-Hölder continuous, thus providing a limitation on the modulus of continuity of the Lyapunov exponent of r…
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。
Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators
针对多频斜移势的一维离散薛定谔算子，证明了在Diophantine频率和大耦合下李雅普诺夫指数一致正且关于能量log-Hölder连续，将结果从单频或简单拟周期模型推广到多频斜移情形。
In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute…
Space-time directional Lyapunov exponents are introduced. They describe the maximal velocity of propagation to the right or to the left of fronts of perturbations in a frame moving with a given velocity. The continuity of these exponents as function of the velocity and an inequality relating them to the directional entropy is proved.
We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.
We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates which lead to optimal Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.
For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e., decompose the set of non wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the conti…
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