Continuous Cocycles Research Articles

STRONG ORBIT EQUIVALENCE OF LOCALLY COMPACT CANTOR MINIMAL SYSTEMS

We study minimal self-homeomorphisms of zero dimensional metrizable locally compact non-compact Hausdorff spaces. For this class of systems, we show that the ordered cohomology group is a complete invariant for strong orbit equivalence, i.e. topological orbit equivalence with continuous orbit cocycles. This is an "infinite" counterpart of a well known result of Giordano, Putnam and Skau about compact Cantor systems.

REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups

In this paper, we prove that any integrable cocycle defined on a “regular” dynamical system, non-atomic and ergodic, and with values in certain connected Lie groups (among which are the connected, abelian or compact, Lie groups) admits a cohomologous continuous cocycle, which generalizes a result of A.V. Koc̆ergin [4] and D.J. Rudolph [6].

Circle Extensions of Z d -Rotations on the d -Dimensional Torus

Let T be an ergodic and free Zdrotation on the d-dimensional torus Td given by T ( m 1 , … , m d ) ( z 1 , … , z d ) = ( e 2 π i ( α 11 m 1 + … + α 1 d m d ) z 1 , … , e 2 π i ( α d 1 m 1 + … + α d d m d ) z d ) , where (m1, …, md) ∈ Zd, (z1, …, zd) ∈ Td and [αjk]j,k=1 …, d ∈ Md(R). For a continuous circle cocycle Φ:Zd × Td → T(Φm+n(z) = Φm(Tnz)Φn(z) for any m, n ∈ Zd), the winding matrix W(Φ) of a cocycle Φ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by TΦ:Zd×Td×T→Td×T, (TΦ)m(z,ω)=(TmZ,Φm(z)ω) are studied. It is shown that if Φ is smooth (for example Φ is of class C1) and det W(Φ) ≠ 0, then TΦ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if Φ is smooth (for example Φ is of class C4), det W(Φ) ≠ 0 and T is a Z2-rotation of finite type, then TΦ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(Φ) = 1, then TΦ has singular spectrum.

Solutions to the twisted cocycle equation over hyperbolic systems

A twisted cocyle with values in a Lie group $G$ is a cocyle that incorporates an automorphism of $G$. Suppose that the underlying transformation is hyperbolic. We prove that if two Holder continuous twisted cocycles with a sufficiently high Holder exponent assign equal 'weights' to the periodic orbits of $\phi$, then they are Holder cohomologous. This generalises a well-known theorem due to Livsic in the untwisted case. Having determined conditions for there to be a solution to the twisted cocycle equation, we consider how many other solution there may be. When $G$ is a toius, we determine conditions for there to be only finitely many solutions to the twisted cocycle equation.

Construction of non-constant and ergodic cocycles

We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general

Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space Θ acting on a Banach space X. Our main tool is the associated evolution semigroup on C0(Θ; X). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mapping/annular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c0(Z; X). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation.

The Spectral Mapping Theorem for Evlution Semigroups on Associated with Strongly Continuous Cocycles

(Theta,mu;X), 1≤p<infinity, associated with a strongly continuous cocycle on a Banach space over a continuous flow on a locally compact metric space Theta.

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

Ergodicity for piecewise smooth cocycles over toral rotations

Let α be an ergodic rotation of the d-torus T = R/Z. For any piecewise smooth function f : T → R with sufficiently regular pieces the unitary operator V h(x) = exp(2πif(x))h(x + α) acting on L2(Td) is shown to have a continuous nonDirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product Sf : Td+1 → Td+1 must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V is singular. In the case d = 1 our technique allows us to extend Pask’s result on ergodicity of cylinder flows on T× R to arbitrary piecewise absolutely continuous real-valued cocycles f satisfying T f = 0 and T f ′ 6= 0. Introduction. Let f be a piecewise absolutely continuous function on the unit interval such that T f ′ 6= 0 and let α ∈ R \Q. In [7] it is shown that the skew product transformation of T2 defined by Sf (x, y) = (x+ α, y + f(x)), where the addition is modulo 1, has a continuous spectrum on the orthocomplement in L2(T2) of the functions depending only on the first variable. Moreover, if f has a single discontinuity with jump c then for every k ∈ Z such that kc is not an integer the spectrum is singular on the invariant subspace Hk = {h(x) exp(2πiky) : h ∈ L2(T)} (see Theorem 2 of [7]; special cases have been proved earlier in [2] and [9]) . By investigating the invariant subspaces Hk the spectral analysis of the skew product reduces to that of the unitary operator V = Vα,f on L2(T), where (V h)(x) = e2πif(x)h(x+ α) (see Anzai [1]), and in fact to the spectral measure σα,f = σ of the function h = 1: 1991 Mathematics Subject Classification: Primary 28D05. Supported in part by KBN grant 2P 03A 07608.

Cohomology of permutative cellular automata

We examineU(d) valued cocycles for a ℤ2+ action generated by a mixing, permutative cellular automaton and show that the set of Holder continuous cocycles (for a given Holder order) which are cohomologous to constant cocycles is both open and closed in the appropriate topology. A continuous dimension function with values in {0, 1,…,d} is defined on cocycles; a cocycle is cohomologous to a constant precisely when the value isd. Whend=1 (the abelian case) the first (essential) cohomology group is countable. IfU(1)≃ circle is replaced by a finite subgroup, this cohomology group is finite.

Remarks on Galois cohomology and definability

In this paper we develop some basic features of Galois cohomology, specifically the connection between first Galois cohomology groups and principal homogeneous spaces, in a model-theoretic context. “Descent theory” also fits into our approach.The model theory involved is elementary, and the reader is referred to [2]. It should be said that we make crucial use of Meq in our analysis. The reader is also referred to Poizat's seminal paper “Une theorie de Galois imaginaire” ([6]). Although our results do not depend on Poizat's work, it is in his paper that the model-theoretic context is suggested for a generalised treatment of Galois theory.Nothing in this paper is particularly deep. We are concerned mainly with translating between the Galois cohomological language and the language of definable sets and definable families of definable sets. We will introduce (in a suitable context) the notion of a definable cocycle (from an automorphism group to a definable group G). The (classical) situation of profinite and continuous cocycles will be a special case. Kolchin's theory of constrained cohomology will be another special case, and our results yield a substantially simpler proof of his Theorem 5 from Chapter VII of [4]. In any case model-theorists will see that definable cocycles correspond to objects with which they are already quite familiar—commuting families of definable bijections.

The Limit Points of Weyl Sums and other Continuous Cocycles

This paper examines the limit points of continuous Z-cocycles defined on a minimal dynamical system and taking values in an abelian metrisable group. This is motivated by and applied to the study of classical exponential series Σ n = 1 N e 2 π i ( n 2 θ + n x ) to show, in particular, that if θ is a transcendental number with lim infqq3/2‖qθ‖ < ∞, then for each x chosen from a dense Gδ subset of [0, 1], these partial sums are dense in C. Indeed the associated cocycle is topologically transitive. Several generalisations to higher degree polynomials and higher dimensions are presented.

On the Structure of Classical and Quantum Flows

In the framework of quantum probability, stochastic flows on manifolds and the interaction representation of quantum physics become unified under the notion ofMarkov cocycle. We prove a structure theorem forσ-weakly continuous Markov cocycles which shows that they are solutions of quantum stochastic differential equations on the largest *-subalgebra, contained in the domain of the generator of the Markov semigroup, canonically associated to the cocycle. The result is applied to prove that any Markov cocycle on the Clifford bundle of a compact Riemannian manifold, whose structure maps preserve the smooth sections and satisfy some natural compatibility conditions, uniquely determines a family of smooth vector fields and a connection, with the property that the cocycle itself is induced by the stochastic flow along the paths of the classical diffusion on the manifold, defined by these vector fields and by the Ito stochastic parallel transport associated to the connection. We use the language and techniques of quantum probability but, even when restricted to the classical case, our results seem to be new.

Absolutely continuous cocycles over irrational rotations

For homeomorphisms $$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$ (z, w ∈S 1,α is irrational,ϕ:S 1→S 1) of the torusS 1×S 1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow

Injury surveillance at the 1985 national Boy Scout Jamboree

We first establish some sufficient conditions for constructing a random exponential attractor for a continuous cocycle on a separable Banach space and weighted spaces of infinite sequences. Then we apply our abstract result to study the existence of random exponential attractors for non-autonomous first order dissipative lattice dynamical systems with multiplicative white noise.

Structure des cocycles markoviens sur l'espace de Fock

We show that strongly continuous unitary Markov cocycles on Fock space are solutions of a quantum stochastic Schrodinger equation and give their explicit form through a decomposition of Fock space on the eigenspaces of the number operator. We also give necessary and sufficient conditions for a generalized Hamiltonian to be the generator of such a cocycle. This generalizes the work of Hudson and Parthasarathy in the norm-continuous case.

On locally continuous cocycles

Cohomology groups of locally continuous cocycles of a topological group G with values in a topological G-module A Ψ are considered and the role of those of degree 2 in the theory of topological extensions of G by A Ψ is analyzed. The exactness of long sequence is proven. It is shown that, if G and A Ψ are Polish and if all topological extensions of G by A Ψ are fibered, the cohomology groups of Borel cocycles and those of locally continuous Borel cocycles are isomorphic for degree 0,1,2.