Rui Zou, Yongluo Cao
We prove a nonuniformly hyperbolic version Livšic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure. Moreover, We also prove a $μ$-continuous solution of the cohomological equation is actually Hölder continuous for the uniform hyperbolic system.
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We consider Holder continuous cocycles over hyperbolic dynamical systems with values in the group of invertible bounded linear operators on a Banach space. We show that two fiber bunched cocycles are Holder continuously cohomologous if and only if they have Holder conjugate periodic data. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base system. We show that this condition can be established based on the periodic data of a cocycle. We also establish Holder continuity of a measurable conjugacy between a fiber bunche…
Versions of the Oseledets multiplicative ergodic theorem for cocycles acting on infinite-dimensional Banach spaces have been investigated since the pioneering work of Ruelle in 1982 and are a topic of continuing research interest. For a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in another requires additional assumptions; a necessary and sufficient condition for the existence of such a dominated splitting was recently given by J. Bochi and N. Gourmelon for invertible finite-dimensional cocycles in discrete time. We extend th…
We consider group-valued cocycles over dynamical systems with hyperbolic behavior. The base system is either a hyperbolic diffeomorphism or a mixing subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We consider the periodic data of $A$, i.e. the set of its return values along the periodic orbits in the base. We show that if the periodic data of $A$ is uniformly quasiconformal or bounded or contained in a compact set, then so is the cocycle. Moreover, in the latter case the cocycle is isometric w…
Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $μ$ with positive entropy, and let $\mathcal{A}$ be a Hölder continuous cocycle of injective bounded linear operators acting on a Banach space $X$. We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\mathcal{A}$ with respect to $μ$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of $\mat…
We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism $f$ of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We prove that upper and lower Lyapunov exponents of $A$ with respect to an ergodic invariant measure $μ$ can be approximated in terms of the norms of the values of $A$ on periodic orbits of $f$. We also show that these exponents cannot always be approximate…
A volume growth-based proof of the Multiplicative Ergodic Theorem for Banach spaces is presented, following the approach of Ruelle for cocycles acting on a Hilbert space. As a consequence, we obtain a volume growth interpretation for the Lyapunov exponents of a Banach space cocycle.
Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces
本文研究了Banach空间上线性余切的Oseledets子空间，在可逆与不可逆情形下证明了其Hölder连续性，推广了已有结果。
The aim of the paper is to present various asymptotic behaviors of skew-evolution semiflows in Banach spaces, as exponential decay, instability, exponential in- stability and integral instability. Relations between these asymptotic properties are also given. As main results, two Datko type theorems are proved. A unified nonuniform approach is provided.
We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by a family of indefinite quadratic Lyapunov-like functionals) and asymptotic compactness. Under such conditions it is possible to reconstruct foliations as in the theory of normally hyperbolic manifolds. Our approach allows to unify many "practical" theories of local and nonlocal invariant manifolds, such as stable/unstable/center manifolds and inertial manifolds, and, moreover, it often leads to op…
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…
Livsic theorem for diffeomorphism cocycles
本研究证明了取值于任意维闭流形的$C^{1+β}$-微分同胚群上闭链的Livšic定理，该结果具有完全全局性，将已有相关结果推广到了更高维度。
We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.
The well-known theorem of Johnson, Palmer and Sell asserts that the endpoints of the Sacker--Sell spectrum of a given cocycle of invertible matrices over a topological dynamical system $(M, f)$ are realized as Lyapunov exponents with respect to some ergodic invariant probability measure for $f$. In this note we establish the version of this result for quasicompact cocycles of operators acting on an arbitrary Banach space.
We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Holder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a contionuous conjugacy between two cocycles.
We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,μ)$ is a non-uniformly hyperbolic system and $A:M \to GL(d,\mathbb{R}) $ is an $α$-Hölder continuous map satisfying $ A(f^{n-1}(p))\ldots A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ and $n\in \mathbb{N}$, there exists a measurable map $P:M\to GL(d,\mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $μ$-almost every $x\in M$. Moreover, we prove that whenever the measure $μ$ has local product structure the transfer map $P$ is $α$-H…
In the present paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.
We deal with the problem of the validity of Livsic's theorem for cocycles of diffeomorphisms satisfying the orbit periodic obstruction over an hyperbolic dynamics. We give a result in the positive direction for cocycles of germs of analytic diffeomorphisms at the origin.
We prove a Livsic type theorem for cocycles taking values in groups of diffeomorphisms of low-dimensional manifolds. The results hold without any localization assumption and in very low regularity. We also obtain a general result (in any dimension) which gives necessary and sufficient conditions to be a coboundary.
We observe that under certain conditions on the Lyapunov exponents a semi-invertible cocycle is, indeed, invertible. As a consequence, if a semi-invertible cocycle generated by a Hölder continuous map $A:M\to M(d, \mathbb{R})$ over a hyperbolic map $f:M\to M$ satisfies a Livšic's type condition, that is, if $A(f^{n-1}(p))\cdot\ldots \cdot A(f(p))A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ then the cocycle is invertible, meaning that $A(x)\in GL(d,\mathbb{R})$ for every $x\in M$, and a Livšic's type theorem is satisfied.
We prove measurable Livsic theorems for dynamical systems modelled by Markov Towers. Our regularity results apply to solutions of cohomological equations posed on Henon-like mappings and a wide variety of nonuniformly hyperbolic systems. We consider both Holder cocycles and cocycles with singularities of prescribed order.
We prove that for semi-invertible and Hölder continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.
A general center manifold theorem on fields of Banach spaces
本文在Banach空间可测场上的非线性余循环中，证明了围绕平稳轨迹的一般局部中心流形定理，推广了中心流形理论的应用范围。
General real measurable Livšic regularity via transfer operators
该研究借助转移算子的抽象假设，证明了非可逆动力系统上实值上循环的一般可测Livšic正则性定理，并将其应用于三类动力系统得到对应正则性结果。
We prove the Livšic Theorem for Hölder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${σ:X\to X}$ that satisfies the Anosov Closing Lemma, and a Hölder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a Hölder continuous transition function if and only if ${a(σ^{n-1}p)\ldots a(σp)a(p)=e}$ for each periodic point $p=σ^n p$.
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with the exponential growth rates of vectors under the action of a linear cocycle on R^d. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of R^d into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recen…
We prove the Livšic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a Hölder continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a Hölder continuous function $C: X \to GL(m,\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on …
We consider linear cocycles over non-uniformly hyperbolic dynamical systems. The base system is a diffeomorphism $f$ of a compact manifold $X$ preserving a hyperbolic ergodic probability measure $μ$. The cocycle $A$ over $f$ is Holder continuous and takes values in $GL(d,R)$ or, more generally, in the group of invertible bounded linear operators on a Banach space. For a $GL(d,R)$-valued cocycle $A$ we prove that the Lyapunov exponents of $A$ with respect to $μ$ can be approximated by the Lyapunov exponents of $A$ with respect to measures on hyperbolic periodic orbits of $f$. In the infinite-di…
We consider the problem of stability and approximability of Oseledets splittings and Lyapunov exponents for Perron-Frobenius operator cocycles associated to random dynamical systems. By developing a random version of the perturbation theory of Gouëzel, Keller, and Liverani, we obtain a general framework for solving such stability problems, which is particularly well adapted to applications to random dynamical systems. We apply our theory to random dynamical systems consisting of $\mathcal{C}^k$ expanding maps on $S^1$ ($k \ge 2$) and provide conditions for the stability of Lyapunov exponents a…
Exponential dichotomy of a strongly continuous cocycle $\bFi$ is proved to be equivalent to existence of a Mañe sequence either for $\bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dynamical spectrum of a product of two cocycles, one of which is scalar, is investigated and applied to describe the essential spectrum of the Euler equation in an arbitrary spacial dimension.
Using multiplicative ergodic theory we prove two formulae describing the relationships between different joint spectral radii for sets of bounded linear operators acting on a Banach space. In particular we recover a formula previously proved by V. S. Shulman and Yu. V. Turovski\uı using operator-theoretic ideas. As a byproduct of our method we answer a question of J. E. Cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle.
We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.
Cocycles in Lie Groups, Cochains and Regularity Problem
本文在Livschitz基础性工作基础上，综述了李群中上循环、上链概念的自然推广及相关正则性结果，涵盖群/半群取值上循环、上同调方程解的正则性传递及高维上同调研究。
The celebrated Livsic theorem states that given M a manifold, a Lie group G, a transitive Anosov diffeomorphism f on M and a Holder function η: M \mapsto G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M \mapsto G satisfying η(x) = ϕ(f(x)) ϕ(x)^{-1} that a condition -- obviously necessary -- on the cocycle generated by ηrestricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications …
Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron-Frobenius operators associated to the maps. Of particular interest is the second-largest Lyapunov exponent, $λ_2$, which can tell us about mixing rates and decay of correlations in the system. We prove a generalized Perron-Frobenius theorem for cocycles of bounded linear operators on Banach spaces that preserve and occasionally contract a cone; this theorem shows that the top Oseledets space for the cocycle i…
The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work represents a significantly simplified means of obtaining the result, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman's subadditive ergodic theorem.
We consider the regularity of measurable solutions $χ$ to the cohomological equation \[ ϕ= χ\circ T -χ, \] where $(T,X,μ)$ is a dynamical system and $ϕ\colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in which $T \colon X\rightarrow X$ is a piecewise $C^k$ Gibbs--Markov map, an affine $β$-transformation of the unit interval or more generally a piecewise $C^{k}$ uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions $χ$ possess $C^k$ versions. In particular we show that if $(T,X,μ)$ is a $β$-transformation then $χ$ has a $C^k$ version, thus…
We consider piecewise cone hyperbolic systems satisfying a bunching condition and we obtain a bound on the essential spectral radius of the associated weighted transfer operators acting on anisotropic Sobolev spaces. The bunching condition is always satisfied in dimension two, and our results give a unifying treatment of the work of Demers-Liverani and our previous work. When the complexity is subexponential, our bound implies a spectral gap for the transfer operator corresponding to the physical measures in many cases (for example if $T$ preserves volume, or if the stable dimension is equal t…
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [GVR].
We present a moduli space for all hyperbolic basic sets of diffeomorphisms on surfaces that have an invariant measure that is absolutely continuous with respect to Hausdorff measure. To do this we introduce two new invariants: the measure solenoid function and the cocycle-gap pair. We extend the eigenvalue formula of A. N. Livsic and Ja. G. Sinai for Anosov diffeomorphisms which preserve an absolutely continuous measure to hyperbolic basic sets on surfaces which possess an invariant measure absolutely continuous with respect to Hausdorff measure. We characterise the Lipschitz conjugacy classes…
How (not) to prove (un)distortion for diffeomorphisms of one-manifolds
研究给定一维流形\(M\)及\(1\leq r\lt s\leq\infty\)，是否存在\(C^s\)保向紧支微分同胚在\(\mathrm{Diff}_{c,+}^s(M)\)中无扭曲但在更大的\(C^r\)微分同胚群中有扭曲。分析了不同\((r,s)\)情况及相关阻碍。
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