Continuous Cocycles Research Articles

The generalised Berger–Wang formula and the spectral radius of linear cocycles

Using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a Banach space. The first formula was previously obtained by V.S. Shulman and Yu.V. Turovskiĭ using operator-theoretic ideas. The second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators. This result may be seen as an extension of classical formulae for the essential spectral radius given by R. Nussbaum, A. Lebow and M. Schechter. Both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system. 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.

Continuous cocycles on locally compact groups: II

In this second paper we continue our development of elementary methods for computing continuous complex-valued solutions of the 2-cocycle functional equation on locally compact groups. The first paper dealt primarily with solvable groups. Here we consider some classical motion groups of mathematical physics, including the Euclidean, Galilean, Lorentz and Poincare groups.

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.

Reducibility of quasiperiodic cocycles in linear Lie groups

Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,ℂ)-reducible is in fact G-reducible modulo two if G=GL(n,ℝ),SL(n,ℝ),Sp(n,ℝ) or O(n) and modulo one if G=U(n) .

Formula omitted]-admissibility and exponential dichotomies of cocycles

We analyze the existence of (eventually no past) exponential and ordinary dichotomies of an exponentially bounded, strongly continuous cocycle { Φ ( θ , t ) } θ ∈ Θ , t ∈ R + over a continuous semiflow σ. Our main tool is the ( L p ( R + , X ) , L q ( R + , X ) ) -admissibility condition (i.e. there exist p , q ∈ [ 1 , ∞ ] , ( p , q ) ≠ ( 1 , ∞ ) , such that for each input f ∈ L p ( R + , X ) and θ ∈ Θ , there exists x ∈ X such that the output u ( ⋅ ; θ , x , f ) : R + → X , u ( t ; θ , x , f ) = Φ ( θ , t ) x + ∫ 0 t Φ ( σ ( θ , s ) , t − s ) f ( s ) d s belongs to L q ( R + , X ) ). We prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to θ. Requiring that the boundedness to be uniform with respect to θ, we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for { Φ ( θ , t ) } θ ∈ Θ , t ∈ R + . Variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained. It is worth to note that we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the well-known concept defined by Sacker and Sell (1994) [33]. Our definition of exponential dichotomy follows partially the definition given in Chow and Leiva (1996) [2] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to Perron (1930) [23], Daleckij and Krein (1974) [6], Massera and Schäffer (1966) [14], Nguyen van Minh, Räbiger and R. Schnaubelt (1998) [18].

Continuous cocycles on locally compact groups

Continuous cocycles on locally compact groups

Compactly supported analytic indices for Lie groupoids

AbstractFor any Lie groupoid we construct an analytic index morphism taking values in a modified K-theory group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constructed in [CR06]. This allows us in particular to prove a more primitive version of the Connes-Skandalis longitudinal index theorem for foliations, that is, an index theorem taking values in a group which pairs with cyclic cocycles. As another application, for D a -PDO elliptic operator with associated index ind we prove that the pairingwith τ a bounded continuous cyclic cocycle, only depends on the principal symbol class [σ(D)]∈K0. The result is completely general for étale groupoids. We discuss some potential applications to the Novikov conjecture.

Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles

AbstractIt is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].

Construction of continuous cocycles for the bicrossed product of locally compact groups

For locally compact groups K, M, and N such that M and N are subgroups of K, K = M ∙ N and M∩N = {e}, where e is the identity of the group K, we give a complete description and propose a method for the construction of pairs of continuous cocycles used in the structure of bicrossed product with cocycles in terms of continuous 2-cocycles on the groups M, N, and K and 3-cocycles on the group K.

On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow

Let $\pi = (\Phi, \sigma)$ be an exponentially bounded, strongly continuous cocycle over a continuous semiflow $\sigma$. We prove that $\pi = (\Phi, \sigma)$ is uniformly exponentially stable if and only if there exist $T>0$ and $c \in(0,1)$, such that for each $\theta \in \Theta$ and $x \in X$ there exists $\tau_{\theta,x} \in (0,T]$ with the property that <p align="center"> $||\Phi(\theta, \tau_{\theta,x})x|| \leq c||x||.$ <p align="left" class="times"> <p align="center"> <p align="left" class="times"> As a consequence of the above result we obtain generalizations, in both continuous-time and discrete-time, of the the well-known theorems of Datko-Pazy, Rolewicz and Zabczyk for an exponentially bounded, strongly continuous cocycle over a semiflow $\sigma$. A version of the above theorems for the case of the exponential instability is also obtained.

Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows

In this paper we investigate the exponential dichotomy of linear skew-product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on Banach bundles. We characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy.

Pullback attractors of nonautonomous reaction–diffusion equations

In this paper, firstly we introduce the concept of norm-to-weak continuous cocycle in Banach space and give a technical method to verify this kind of continuity, then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle, using the measure of noncompactness. As an application, we prove the existence of pullback attractors in H 0 1 of the cocycle associated with the solutions for some nonlinear nonautonomous reaction–diffusion equations. The attractor pullback attracts all bounded subsets of H 0 1 in the norm of H 0 1 .

Cocycles and Mañe sequences with an application to ideal fluids

Exponential dichotomy of a strongly continuous cocycle Φ is proved to be equivalent to existence of a Mañe sequence either for Φ 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.

The Lyapunov exponents of generic volume-preserving and symplectic maps

We show that the integrated Lyapunov exponents of C1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents are equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere. We deduce a sharp dichotomy for generic volume-preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero. Similarly, for a residual subset of all C1 symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2. Finally, given any set S ?? GL(d) satisfying an accessibility condition, for a residual subset of all continuous S-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on S is satisfied for most common matrix groups and also for matrices that arise from discrete Schr?Nodinger operators.

Heat Kernels on homogeneous spaces

AbstractLet a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

Novikov-Morse theory for dynamical systems

The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley's theory for dynamical systems. We introduce the concept of a flow carrying a cocycle \(\alpha\), (generalized) \(\alpha\)-flow for short, where \(\alpha\) is a continuous cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define \(\alpha\)-Morse-Smale flows that allow the existence of “cycles” in contrast to the usual Morse-Smale flows. \(\alpha\)-flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle \(\alpha\), we construct a so-called \(\pi\)-Morse decomposition for an \(\alpha\)-flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [12]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when \(\alpha\) is a coboundary) as well as the Novikov type inequalities ( when \(\alpha\) is a nontrivial cocycle).

Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices

We propose a geometric sufficient criterium “à la Furstenberg” for the existence of non-zero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: non-existence of probability measures on the fibers invariant under the cocycle and under the holonomies of the stable and unstable foliations of the transformation. This criterium applies to locally constant and to dominated cocycles over hyperbolic sets endowed with an equilibrium state.As a consequence, we get that non-zero exponents exist for an open dense subset of these cocycles, which is also of full Lebesgue measure in parameter space for generic parametrized families of cocycles.This criterium extends to continuous time cocycles obtained by lifting a hyperbolic flow to a projective fiber bundle, tangent to some foliation transverse to the fibers. Again, non-zero Lyapunov exponents are implied by non-existence of transverse measures invariant under the holonomy of the foliation.We apply this last result to a natural geometric context: the geodesic flow tangent to the leaves of a foliation obtained as the suspension of a representation ρ:π1(S)→PSL(2,ℂ) of the fundamental group of a hyperbolic compact surface. We prove the existence of non-zero Lyapunov exponents for the corresponding cocycle, for ρ in a dense open subset of all the representations. As a consequence we get that this foliated geodesic flow has a unique Sinai–Ruelle–Bowen measure.

Uniform (projective) hyperbolicity or no hyperbolicity: A dichotomy for generic conservative maps

We show that the Lyapunov exponents of volume preserving C1 diffeomorphisms of a compact manifold are continuous at a given diffeomorphism if and only if the Oseledets splitting is either dominated or trivial. It follows that for a C1-residual subset of volume preserving diffeomorphisms the Oseledets splitting is either dominated or trivial.We obtain analogous results in the setting of symplectic diffeomorphisms, where the conclusion is actually stronger: dominated splitting is replaced by partial hyperbolicity. We also obtain versions of these results for continuous cocycles with values in some matrix groups.In the text we give the precise statements of these results and the ideas of the proofs. The complete proofs will appear in [4].

Fundamental cocycles of tiling spaces

We study continuous cocycles defined on the set of planar tilings with values in discrete groups. Following Schmidt we show that for generalized domino tilings, L-tiles, and some systems of paths there exists a fundamental cocycle, i. e. we find a cocycle cf , so that all other continuous cocycles c are cohomologous to a homomorphic image of cf .