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 α, (generalized) α-flow for short, where α is a cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characteriz...

We show that the Seiberg-Witten map for a noncommutative gauge theory involves a noncommutative 1-cocycle. The cocycle condition enforces a consistency requirement, which has been previously derived.

Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ). A more compact way of saying this is to assert that such a cocycle is a map of simplicial sheaves C(U)→ BG on th...

We present a general setting to investigate Ufin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some Ufin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. R. Parthasarathy and K. Schmidt to give a necessary cohomological condition on a gr...

The paper [FGTQ14] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations, including “Ulam’s method”, a popular numerical method for approximating acims. The expansivity requirements of [FGTQ14] were that the cocycle (or powers of the cocycle) should be “expanding on average” before app...

Abstract. This paper concerns the structure of the space C of real valued cocycles for a flow (X,Zm). We show that C is always larger than the set of cocycles cohomologous to the linear maps if the flow has a free dense orbit. By considering appropriate dual spaces for C, we obtain the concept of an invariant cocycle integral. The extreme points of the set of invariant cocycle integrals paralle...

K. Cho and S. Nelson introduced the notion of a quandle coloring quiver, which is quiver-valued link invariant, cocycle quiver an enhancement by assigning to each vertex weight computed using 2-cocycle. In this paper, we study quivers dihedral quandles introduce notions shadow are versions quiver. We show that, when use prime order, equivalent numbers order Mochizuki's 3-cocycle, invariants.