Continuous Cocycles Research Articles

Extending Some Results of L. Barreira and C. Valls to the Case of Linear Skew-Product Semiflows

In a paper from 2010, Barreira and Valls (J Differ Equ 249:2889–2904, 2010) use Lyapunov norms to set up admissibility conditions for nonuniform exponential contractions. In 2011, the same authors extend their analysis from Barreira and Valls (J Differ Equ 249:2889–2904, 2010) to the case of nonuniform exponential dichotomy for linear evolution families with (non)uniform exponential growth (see Barreira and Valls in Discret Contin Dyn Syst 30(1):39–53, 2011). Following their approach, we are able to choose appropriate “test-functions” to establish admissibility-type conditions for the existence of a (non)uniform exponential dichotomy of a strongly continuous cocycle with (non)uniform exponential growth.

Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped sine–Gordon equation on unbounded domains

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped sine–Gordon equation driven by multiplicative noise defined on an unbounded domain. 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.

The cocycle identity holds under Stopping

In recent work of the authors, it was shown how to use any finite quantum stop time to stop the CCR flow and its strongly continuous isometric cocycles (Q. J. Math. 65:1145–1164, 2014). The stopped cocycle was shown to satisfy a stopped form of the cocycle identity, valid for deterministic increments of the time used for stopping. Here, a generalization of this identity is obtained, where both cocycle parameters are replaced with finite quantum stop times.

Nonexistence of Lyapunov exponents for matrix cocycles

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_\delta$ set).

PULLBACK ATTRACTORS FOR MODIFIED SWIFT-HOHENBERG EQUATION ON UNBOUNDED DOMAINS WITH NON-AUTONOMOUS DETERMINISTIC AND STOCHASTIC FORCING TERMS

In this paper, the existence and uniqueness of pullback attractors for the modified Swift-Hohenberg equation defined on Rn driven by both deterministic non-autonomous forcing and additive white noise are established. We first define a continuous cocycle for the equation in L2(Rn), and we prove the existence of pullback absorbing sets and the pullback asymptotic compactness of solutions when the equation with exponential growth of the external force. The long time behaviors are discussed to explain the corresponding physical phenomenon.

Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.

Fiber bunching and cohomology for Banach cocycles over hyperbolic systems

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 bunched cocycle and one with values in a set which is compact in strong operator topology.

Livšic theorem for banach rings

The Livsic Theorem for Holder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Holder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Holder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.

EQUIVARIANT GEOMETRY OF BANACH SPACES AND TOPOLOGICAL GROUPS

We study uniform and coarse embeddings between Banach spaces and topological groups. A particular focus is put on equivariant embeddings, that is, continuous cocycles associated to continuous affine isometric actions of topological groups on separable Banach spaces with varying geometry.

Continuous cocycle superrigidity for shifts and groups with one end

We prove that if a finitely generated group G is not torsion then a necessary and sufficient condition for every full shift over G has (continuous) cocycle superrigidity is that G has one end. It is a topological version of the well-known Popa’s measurable cocycle superrigidity theorem (Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math. 170(2):243–295, 2007). For the proof of sufficiency, we introduce a new specification property for shifts over general groups which plays a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method of using homoclinic equivalence relation that was introduced by Schmidt (Pac J Math 170(1):237–269, 1995) to study cocycle rigidity for $${{\mathbb {Z}}}^d$$ -shifts. Indeed, in this direction we prove this superrigidity result for certain more general systems. And for the necessary condition, we apply Specker’s characterization for ends of groups via the associated first cohomology groups to get the result. Finally, combining our results with Li (Ergod Theory Dyn Syst. arXiv:1503.01704 , Theorem 1.6), we have an application in continuous orbit equivalence rigidity.

Continuous orbit equivalence rigidity

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of$C^{\ast }$-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.

Limit algebras and integer-valued cocycles, revisited

A triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C*-correspondence if and only if its fundamental relation R(A) is a tree admitting a $Z^+_0$-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C*-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory.

Discontinuity of Lyapunov exponents near fiber bunched cocycles

We give examples of locally constant $\text{SL}(2,\mathbb{R})$-cocycles over a Bernoulli shift that are discontinuity points for Lyapunov exponents in the Hölder topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author [Continuity of Lyapunov exponents for cocycles with invariant holonomies. Preprint, 2015, arXiv:1507.08978] have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.

Hölder continuity of Oseledets splittings for semi-invertible operator cocycles

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc.93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a non-invertible cocycle with non-vanishing Lyapunov exponents exhibits non-uniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.

RANDOM ATTRACTOR FOR NON-AUTONOMOUS STOCHASTIC STRONGLY DAMPED WAVE EQUATION ON UNBOUNDED DOMAINS

In this paper we study the asymptotic dynamics for the nonautonomous stochastic strongly damped wave equation driven by additive noise defined on unbounded domains. First we introduce a continuous cocycle for the equation and then investigate the existence and uniqueness of tempered random attractors which pullback attract all tempered random sets.

Complex one-frequency cocycles

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial . The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

Hyperbolic groupoids: metric and measure

We construct Patterson–Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces and Anosov flows using Gromov hyperbolic graphs. Other results include a duality theory for hyperbolic groupoids graded by Hölder continuous Busemann cocycles, and a characterization of visual metrics.

Cocycles with one exponent over partially hyperbolic systems

We consider Hölder continuous linear cocycles over partially hyperbolic diffeomorphisms. For fiber bunched cocycles with one Lyapunov exponent we show continuity of measurable invariant conformal structures and sub-bundles. Further, we establish a continuous version of Zimmer’s Amenable Reduction Theorem. For cocycles over hyperbolic systems we also obtain polynomial growth estimates for the norm and the quasiconformal distortion from the periodic data.

Continuity properties of measurable group cohomology

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to `regularize' measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fr\'echet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fr\'echet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣σ) for some 0 < σ < 1.