Space Of Cocycles Research Articles

Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.

Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams

We introduce a differential refinement of Cohomotopy cohomology theory, defined on Penrose diagram spacetimes, whose cocycle spaces are unordered configuration spaces of points. First we prove that brane charge quantization in this differential 4-Cohomotopy theory implies intersecting p/(p+2)-brane moduli given by ordered configurations of points in the transversal 3-space. Then we show that the higher (co-)observables on these brane moduli, conceived as the (co-)homology of the Cohomotopy cocycle space, are given by weight systems on horizontal chord diagrams and reflect a multitude of effects expected in the microscopic quantum theory of Dp/D(p+2)-brane intersections: condensation to stacks of coincident branes and their Chan-Paton factors, BMN matrix model and fuzzy funnel states, M2-brane 3-algebras, the Hanany-Witten rules, AdS3-gravity observables, supersymmetric indices of Coulomb branches as well as gauge/gravity duality between all these. We discuss this in the context of the hypothesis that the M-theory C-field is charge-quantized in Cohomotopy theory.

Clean tangled clutters, simplices, and projective geometries

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.Let C be a clean tangled clutter. It was recently proved that C has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of C. The core is a duplication of the cuboid of a set of 0−1 points, called the setcore of C.In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then C has the clutter of the lines of the Fano plane as a minor.Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.

From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups

We study a family of functions defined in a very simple way as sums of powers of binary Hermitian forms with coefficients in the ring of integers of an Euclidean imaginary quadratic field K with discriminant dK. Using these functions we construct a nontrivial cocycle belonging to the space of parabolic cocycles on Euclidean Bianchi groups. We also show that the average value of these functions is related to the special values of L(χdK,s). Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L(χdK,s).

On the spectral radius of compact operator cocycles

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger–Wang’s formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle.

Hyperreflexivity of the bounded $n$-cocycle spaces of Banach algebras with matrix representations

Hyperreflexivity of the bounded $n$-cocycle spaces of Banach algebras with matrix representations

A control theorem for p-adic automorphic forms and Teitelbaum’s $$\mathcal {L}$$-invariant

In this article, we describe an efficient method for computing Teitelbaum's $p$-adic $\mathcal{L}$-invariant. These invariants are realized as the eigenvalues of the $\mathcal{L}$-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree $\mathcal{T}$, which is computable by the methods of Franc and Masdeu described in [FM14]. The main difficulty in computing the $\mathcal{L}$-operator is the efficient computation of the $p$-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In order to make these methods applicable to our setting, we prove a control theorem for $p$-adic automorphic forms of arbitrary even weight. Moreover, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants of different levels and weights for $p=2$.

Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We establish large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we obtain continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem for the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible everywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

Livšic theorem for diffeomorphism cocycles

We prove the so called Liv\v{s}ic theorem for cocycles taking values in the group of $C^{1+\beta}-diffeomorphisms of any closed manifold of arbitrary dimension. Since no localization hypothesis is assumed, this result is completely global in the space of cocycles and thus extends the previous result of the second author and Potrie [KP16] to higher dimensions.

Group 1-cohomology is complemented

We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,\pi)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.

An approach to intersection theory on singular varieties using motivic complexes

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky–MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based on generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

Cycle/Cocycle Oblique Projections on Oriented Graphs

It is well known that the edge vector space of an oriented graph can be decomposed in terms of cycles and cocycles (alias cuts, or bonds), and that a basis for the cycle and the cocycle spaces can be generated by adding and removing edges to an arbitrarily chosen spanning tree. In this paper we show that the edge vector space can also be decomposed in terms of cycles and the generating edges of cocycles (called cochords), or of cocycles and the generating edges of cycles (called chords). From this observation follows a construction in terms of oblique complementary projection operators. We employ this algebraic construction to prove several properties of unweighted Kirchhoff-Symanzik matrices, encoding the mutual superposition between cycles and cocycles. In particular, we prove that dual matrices of planar graphs have the same spectrum (up to multiplicities). We briefly comment on how this construction provides a refined formalization of Kirchhoff's mesh analysis of electrical circuits, which has lately been applied to generic thermodynamic networks.

Hyperreflexivity of bounded $$N$$ N -cocycle spaces of banach algebras

Let \(X\) and \(Y\) be Banach spaces, \(n\in \mathbb {N}\), and \(B^n(X,Y)\) the space of bounded \(n\)-linear maps from \(X\times \ldots \times X\) (\(n\)-times) into \(Y\). The concept of hyperreflexivity has already been defined for subspaces of \(B(X,Y)\), where \(X\) and \(Y\) are Banach spaces. We extend this concept to the subspaces of \(B^n(X,Y)\), taking into account its \(n\)-linear structure. We then investigate when \(\mathcal {Z}^n(A,X)\), the space of all bounded \(n\)-cocycles from a Banach algebra \(A\) into a Banach \(A\)-bimodule \(X\), is hyperreflexive. Our approach is based on defining two notions related to a Banach algebra, namely the strong property \((\mathbb {B})\) and bounded local units, and then applying them to find uniform criterions under which \(\mathcal {Z}^n(A,X)\) is hyperreflexive. We also demonstrate that these criterions are satisfied in variety of examples including large classes of C\(^*\)-algebras and group algebras and thereby providing various examples of hyperreflexive \(n\)-cocyle spaces. One advantage of our approach is that not only we obtain the hyperreflexivity for bounded \(n\)-cocycle spaces in different cases but also our results generalize the earlier ones on the hyperreflexivity of bounded derivation spaces, i.e. when \(n=1\), in the literature. Finally, we investigate the hereditary properties of the strong property \((\mathbb {B})\) and b.l.u. This allows us to come with more examples of bounded \(n\)-cocycle spaces which are hyperreflexive.

Properties of Gomory–Hu co-cycle bases

Let D = ( V , A ) be a directed graph with non-negative arc weights. We study the problem of computing certain special co-cycle bases of D , in particular, a minimum weight weakly fundamental co-cycle basis. A co-cycle in D corresponds to a cut in the underlying undirected graph and a { − 1 , 0 , 1 } arc incidence vector is associated with each co-cycle, where the ± 1 coordinates are used for the arcs crossing the cut. The weight of a co-cycle C is the sum of the weights of those arcs a such that C ( a ) = ± 1 . The vector space over Q generated by the arc incidence vectors of the co-cycles is the co-cycle space of D . The co-cycle space of D can also be defined as the orthogonal complement of the cycle space of D . A set of linearly independent co-cycles that span the co-cycle space is a co-cycle basis of D . The problem of computing a co-cycle basis { C 1 , … , C k } such that the sum of weights of the co-cycles in the basis is the least possible is the minimum co-cycle basis problem. A co-cycle basis { C 1 , … , C k } is weakly fundamental if for every i there is an arc a i such that C i ( a i ) = ± 1 while C j ( a i ) = 0 for j > i . The minimum cycle basis problem in directed and undirected graphs is a well-studied problem and while polynomial time algorithms are known for these problems, the problem of computing a minimum weight weakly fundamental cycle basis has recently been shown to be APX-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory–Hu tree T of the underlying undirected graph of D is a minimum weight weakly fundamental co-cycle basis of D . This is, in fact, a minimum co-cycle basis of D and it is also totally unimodular. Thus this is a special co-cycle basis that simultaneously answers several questions in the domain of co-cycle bases. It is known that there is no such special cycle basis for general graphs.

Euclidian embeddings of periodic nets: definition of a topologically induced complete set of geometric descriptors for crystal structures

Crystal-structure topologies, represented by periodic nets, are described by labelled quotient graphs (or voltage graphs). Because the edge space of a finite graph is the direct sum of its cycle and co-cycle spaces, a Euclidian representation of the derived periodic net is provided by mapping a basis of the cycle and co-cycle spaces to a set of real vectors. The mapping is consistent if every cycle of the basis is mapped on its own net voltage. The sum of all outgoing edges at every vertex may be chosen as a generating set of the co-cycle space. The embedding maps the cycle space onto the lattice L. By analogy, the concept of the co-lattice L* is defined as the image of the generators of the co-cycle space; a co-lattice vector is proportional to the distance vector between an atom and the centre of gravity of its neighbours. The pair (L, L*) forms a complete geometric descriptor of the embedding, generalizing the concept of barycentric embedding. An algebraic expression permits the direct calculation of fractional coordinates. Non-zero co-lattice vectors allow nets with collisions, displacive transitions etc. to be dealt with. The method applies to nets of any periodicity and dimension, be they crystallographic nets or not. Examples are analyzed: α-cristobalite, the seven unstable 3-periodic minimal nets etc.

Another Characterisation of Planar Graphs

A new characterisation of planar graphs is presented. It concerns the structure of the cocycle space of a graph, and is motivated by consideration of the dual of an elementary property enjoyed by sets of circuits in any graph.

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dimA(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dimA(M)=r(M). In 2004, when |A|=1, Lemos proved that dimA(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dimA(M)⩾r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.

A POLYNOMIAL APPROACH TO COCYCLES OVER ELEMENTARY ABELIAN GROUPS

AbstractWe derive bivariate polynomial formulae for cocycles and coboundaries in Z2(ℤpn,ℤpn), and a basis for the (pn−1−n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the $(2^n + {n\choose 2} -1)$-dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form.

AN OPEN SET OF UNBOUNDED COCYCLES WITH SIMPLE LYAPUNOV SPECTRUM AND NO EXPONENTIAL SEPARATION

We give an example of an open set of unbounded cocycles satisfying the integrability condition of the multiplicative ergodic theorem such that all the cocycles in this open set have simple Lyapunov spectrum but have no exponential separation. Thus, unlike the bounded case, the exponential separation property is nongeneric in the space of unbounded cocycles.

The Weil-Petersson and Thurston symplectic forms

We consider the Weil-Petersson form on the Teichmüller space $\mathscr {T}$(S) of a surface S of genus at least 2, and we compute it in terms of the shearing coordinates for $\mathscr {T}$(S) associated to a geodesic lamination λ on S. In the corresponding expression, the Weil-Petersson form coincides with Thurston's intersection form on the space of transverse cocycles for λ.