Continuous Cocycles Research Articles

Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

Consider a compact group $G$ acting on a real or complex Banach Lie group $U$, by automorphisms in the relevant category, and leaving a central subgroup $K\le U$ invariant. We define the spaces ${}_KZ^n(G,U)$ of $K$-relative continuous cocycles as those maps ${G^n\to U}$ whose coboundary is a $K$-valued $(n+1)$-cocycle; this applies to possibly non-abelian $U$, in which case $n=1$. We show that the ${}_KZ^n(G,U)$ are analytic submanifolds of the spaces $C(G^n,U)$ of continuous maps $G^n\to U$ and that they decompose as disjoint unions of fiber bundles over manifolds of $K$-valued cocycles. Applications include: (a) the fact that ${Z^n(G,U)\subset C(G^n,U)}$ is an analytic submanifold and its orbits under the adjoint of the group of $U$-valued $(n-1)$-cochains are open; (b) hence the cohomology spaces $H^n(G,U)$ are discrete; (c) for unital $C^*$-algebras $A$ and $B$ with $A$ finite-dimensional the space of morphisms $A\to B$ is an analytic manifold and nearby morphisms are conjugate under the unitary group $U(B)$; (d) the same goes for $A$ and $B$ Banach, with $A$ finite-dimensional and semisimple; (e) and for spaces of projective representations of compact groups in arbitrary $C^*$ algebras (the last recovering a result of Martin's).

Asymptotic behavior for stochastic plate equations on unbounded domains

In this paper, we investigate the long-time behavior of the solutions for stochastic plate equations with memory and additive noise. First we establish a continuous cocycle for the equation and the pullback asymptotic compactness of solutions. Second we prove the existence of random attractors for the equation.

A note on the marginal instability rates of two-dimensional linear cocycles

A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant -cocycle on the full shift is not exponential, then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.

Continuous orbit equivalence rigidity for left-right wreath product actions

Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung.

Asymptotic behavior of plate equations driven by colored noise on unbounded domains

This paper investigates mainly the asymptotic behavior of the nonautonomous random dynamical systems generated by the plate equations driven by colored noise defined on mathbb{R}^{n}. First, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

AbstractWe extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$ . 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 $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $

<p style='text-indent:20px;'>This paper deals with the asymptotic behavior of the solutions to a class of non-autonomous <i>fractional</i> stochastic <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation driven by linear additive noise on the entire space <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We firstly prove the existence of a continuous non-autonomous cocycle for the equation as well as the uniform estimates of solutions. We then show pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors and the uniform tail-estimates of the solutions for large space variables when time is large enough to surmount the lack of compact Sobolev embeddings on unbounded domains. Finally, we establish the upper semi-continuity of the random attractors when noise intensity approaches zero.</p>

Asymptotic Behavior of Random Reaction-diffusion Delay Equations Driven by Colored Noise

In this paper, we consider the asymptotic behavior of random reaction-diffusion delay equations driven by colored noise defined on unbounded domains. We firstly establish the existence and uniqueness of pullback random attractors for the continuous cocycle associated with the equation, and then consider the convergence of random attractors when colored noise approximates white noise. The methods of uniform tail-estimate and operator decomposition are employed to obtain the pullback asymptotic compactness of the solutions in order to overcome the non-compactness of the Sobolev embeddings on unbounded domains.

Hölder continuity of Oseledets subspaces for linear cocycles on Banach spaces* *This work is partially supported by NSFC (12271386, 11871361), and the second author is partially supported by Qinglan project from Jiangsu Province.

Let f : X → X be an invertible Lipschitz transformation on a compact metric space X. Given a Hölder continuous invertible operator cocycles on a Banach space and an f-invariant ergodic measure, this paper establishes the Hölder continuity of Oseledets subspaces over a compact set of arbitrarily large measure. This extends a result in [V Araujo, A I Bufetov and S Filip, On Hölder-continuity of Oseledets subspaces, J. Lond. Math. Soc. 2016, 93 : 194–218]. for invertible operator cocycles on a Banach space. This paper also proves the Hölder continuity in the non-invertible case. Finally, some applications are given in the end of this paper.

Random attractors for non-autonomous stochastic Brinkman-Forchheimer equations on unbounded domains

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of the non-autono-mous stochastic 3D Brinkman-Forchheimer equations on unbounded domains. We first define a continuous non-autonomous cocycle for the stochastic equations, and then prove that the existence of tempered random attractors by Ball's idea of energy equations. Furthermore, we obtain that the tempered random attractors are periodic when the deterministic non-autonomous external term is periodic in time.</p>

Stochastic dynamics of non-autonomous fractional Ginzburg-Landau equations on $ \mathbb{R}^3 $

<p style='text-indent:20px;'>We investigate a class of non-autonomous non-local fractional stochastic Ginzburg-Landau equation with multiplicative white noise in three spatial dimensions. Of particular interest is the asymptotic behavior of its solutions. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in <inline-formula><tex-math id="M2">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula>. The existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions is then established. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in <inline-formula><tex-math id="M3">\begin{document}$ L^2( \mathbb{R}^3) $\end{document}</tex-math></inline-formula> is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.</p>

Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains

<p style='text-indent:20px;'>In this paper we study asymptotic behavior of a class of stochastic plate equations with memory and additive noise. First we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions. Second we consider the existence and upper semicontinuity of random attractors for the equation.</p>

Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

<abstract><p>The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.</p></abstract>

Continuity of sub-additive topological pressure with matrix cocycles * *The first author is partially supported by NSFC (11901305), the second author is partially supported by NSFC (11771317, 11790274), the third author is partially supported by NSFC (11871361, 11790274) and Qinglan project from Jiangsu Province.

Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.

Typical properties of periodic Teichmüller geodesics: Lyapunov exponents

AbstractConsider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

Random exponential attractor for a non-autonomous Zakharov lattice system with multiplicative white noise

We prove the existence of a random exponential attractor for a non-autonomous Zakharov lattice system with multiplicative white noise. We first transfer the stochastic lattice system into an equivalent random lattice system whose solutions generate a continuous cocycle on a phase space of infinite sequences. Then we estimate the tail of solutions for the random system. We next verify that the continuous cocycle satisfies the Lipschitz continuity and the random squeezing property on a tempered random closed absorbing set. Finally, we check the boundedness of the expectation of some random variables and obtain the existence of a random exponential attractor.

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

AbstractWe consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

Groupoid algebras as covariance algebras

Suppose G is a second-countable locally compact Hausdorff \'{e}tale groupoid, G is a discrete group containing a unital subsemigroup P, and c:G→G is a continuous cocycle. We derive conditions on the cocycle such that the reduced groupoid C∗-algebra C∗r(G) may be realised as the covariance algebra of a product system over P with coefficient algebra C∗r(c−1(e)). When (G,P) is a quasi-lattice ordered group, we also derive conditions that allow C∗r(G) to be realised as the Cuntz--Nica--Pimsner algebra of a compactly aligned product system.

Existence of attractors for stochastic diffusion equations with fractional damping and time-varying delay

This paper deals with the well-posedness and existence of attractors of a class of stochastic diffusion equations with fractional damping and time-varying delay on unbounded domains. We first prove the well-posedness and the existence of a continuous non-autonomous cocycle for the equations and the uniform estimates of solutions and the derivative of the solution operators with respect to the time-varying delay. We then show pullback asymptotic compactness of solutions and the existence of random attractors by utilizing the Arzelà–Ascoli theorem and the uniform estimates for the derivative of the solution operator in the fractional Sobolev space Hα(Rn), with 0 < α < 1.

Pullback random attractors of stochastic strongly damped wave equations with variable delays on unbounded domains

<abstract><p>In this paper, we consider the asymptotic behavior of solutions to stochastic strongly damped wave equations with variable delays on unbounded domains, which is driven by both additive noise and deterministic non-autonomous forcing. We first establish a continuous cocycle for the equations. Then we prove asymptotic compactness of the cocycle by tail-estimates and a decomposition technique of solutions. Finally, we obtain the existence of a tempered pullback random attractor.</p></abstract>