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In this paper, we generalize the Avalanche Principle in [Ann. of. Math. (2), 2001, 154(1): 155–203]. [ABSTRACT FROM AUTHOR]

We present a decomposition of rational twisted G-equivariant K-theory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [J. Geom. Phys. 6 (1989), pp. 671–677] as well as the decomposition by Adem and Ruan [Comm. Math. Phys. 237 (2003), pp. 533–556] for twists coming from group cocycles. [ABSTRACT FROM AUTHOR]

Let G be a split, simple, simply connected, algebraic group over Q . The degree 4, weight 2 motivic cohomology group of the classifying space BG of G is identified with Z . We construct cocycles representing the generator, known as the second universal motivic Chern class. If G = SL (m) , there is a canonical cocycle, defined by Goncharov (Explicit construction of characteristic classes. Advances in Soviet mathematics, 16, vol 1. Special volume dedicated to I.M.Gelfand's 80th birthday, pp 169–210, 1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of G -orbits on the cube of the principal affine space G / U . Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides the motivic generator of H 3 (G (C) , Z (2)) . However to lift it to the whole cocycle we need cluster coordinates: construction of the first two components uses crucially the cluster structure of the moduli spaces A (G , S) related to the moduli space of G -local systems on S . In retrospect, it partially explains why cluster coordinates on the space A (G , S) should exist. The construction has numerous applications, including explicit constructions of the universal extension of the group G by K 2 , the line bundle on Bun (G) generating its Picard group, Kac–Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a G -bundle. It is a motivic analog of the work of Gabrielov et al. (1974), for any G . We show that the cluster construction of the measurable group 3-cocycle for the group G (C) , provided by our motivic cocycle, gives rise to the quantum deformation of its exponent. [ABSTRACT FROM AUTHOR]

The paper presents some characterizations for the study of dichotomy of dynamical systems. Integral characterizations of Datko type are considered for the notion of uniform dichotomy with growth rates, also called uniform h-dichotomy, for skew-evolution cocycles in Banach spaces.

Abstract This paper concerns the long term behavior of the stochastic two-dimensional g-Navier–Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain.

When analytic quasi-periodic cocycles are defined on a high dimension torus, their Lyapunov exponents have perturbative positivity and continuity. In this article, we study a class of analytic quasi-periodic Jacobi cocycles defined on a two dimension torus. We show that in the non-perturbative large coupling regimes, the Lyapunov exponent is positive for any frequency and weak Holder continuous for the full-measured frequency.

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Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh and Sibony (Ann. Sci. Ec. Norm. Supér. (4) 37 (2004) 959-971), and by Truong (J. Reine Angew. Math. 758 (2020) 139-182). Precisely, we give a new proof of the submultiplicativity properties of these degrees and of their birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao (Ann. Inst. Fourier (Grenoble) 65 (2015) 1367-1369) and Popovici (Math. Ann. 364 (2016) 649-655), which generalizes Siu's inequality (see Trapani, Math. Z 219 (1995) 387-401) to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in X. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

We wish to point out errors in the paper "Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers", International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the "ramified case" of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the (S,n)-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected. Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let I be an ideal such that In is principal in OF,S⁠. Write In = (ƒ−1)⁠. Then the Kummer cocycles kn(ƒ) will be in Z1(U, Z/nZ)⁠. For any a ∈ F⁠ , denote by aS its image in ∏v∈S Fv⁠. Thus, we get an element   [ƒ]S,n := [(kn(&#402), kn2(ƒS), 0)] ∈ Z1(U, Z/nZ × SZ/n2Z), which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both μn2 and μn⁠.) [ABSTRACT FROM AUTHOR]

It is presented and proved a version of Livschitz Theorem for hyperbolic flows pragmatically oriented to the cohomological context. Previously, it is introduced the concept of cocycle and a natural notion of symmetry for cocycles. It is discussed the fundamental relationship between the existence of solutions of cohomological equations and the behavior of the cocycles along periodic orbits. The generalization of this theorem to a class of suspension flows is also discussed and proved. This generalization allows giving a different proof of the Livschitz Theorem for flows based on the construction of Markov systems for hyperbolic flows.

We consider the quasi-periodic cocycles $(\omega,A(x,E)): (x,v)\mapsto (x+\omega, A(x,E)v)$ with $\omega$ Diophantine. Let $M_2(\mathbb{C})$ be a normed space endowed with the matrix norm, whose elements are the $2\times 2$ matrices. Assume that $A:\mathbb{T}\times \mathscr{E}\to M_2(\mathbb{C})$ is jointly continuous, depends analytically on $x\in\mathbb{T}$ and is Holder continuous in $E\in\mathscr{E}$, where $\mathscr{E}$ is a compact metric space and $\mathbb{T}$ is the torus. We prove that if two Lyapunov exponents are distinct at one point $E_0\in\mathscr{E}$, then these two Lyapunov exponents are Holder continuous at any E in a ball central at $E_0$. Moreover, we will give the expressions of the radius of this ball and the Holder exponents of the two Lyapunov exponents.

We give an overview of different characterisations of L∞-structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L∞-algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L∞-algebras by 2-cocycles.

In this article, we study the uniform asymptotic stability of the switched system $u'=f_{ u(t)}(u)$, $uin mathbb{R}^n$, where $ u:mathbb{R}_{+}o {1,2,dots,m}$ is an arbitrary piecewise constant function. We find criteria for the asymptotic stability of nonlinear systems. In particular, for slow and homogeneous systems, we prove that the asymptotic stability of each individual equation $u'=f_p(u)$ ($pin {1,2,dots,m}$) implies the uniform asymptotic stability of the system (with respect to switched signals). For linear switched systems (i.e., $f_p(u)=A_pu$, where $A_p$ is a linear mapping acting on $E^n$) we establish the following result: The linear switched system is uniformly asymptotically stable if it does not admit nontrivial bounded full trajectories and at least one of the equations $x'=A_px$ is asymptotically stable. We study this problem in the framework of linear non-autonomous dynamical systems (cocyles).

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial π2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group acting ergodically on X. We discuss relationships between homomorphisms on the measured groupoid X×G, conjugacy of skew product extensions, and similarity of measured groupoids. To do this, we describe the structure of homomorphisms on X×G whose restriction to an extension given by a skew product action is the trivial homomorphism.

In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy.

We identify Čech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L ∞-algebra. Applications to deformation theory are described. [ABSTRACT FROM PUBLISHER]

We prove that all strongly outer -actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable -actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other. [ABSTRACT FROM AUTHOR]

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings. [ABSTRACT FROM AUTHOR]

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic probability which is positive on non-empty open sets, we conclude that there is a C0-residual subset of cocycles within which, for almost every x, either the Oseledets–Ruelle's decomposition along the orbit of x is dominated or all the Lyapunov exponents are equal to -∞. [ABSTRACT FROM AUTHOR]

We introduce a new type of generalized derivations associate with Hochschild 2-cocycles and prove that every generalized Jordan derivation of this type is a generalized derivation under certain conditions. This result contains the results of I. N. Herstein [6, Theorem 3.1] and M. Ashraf and N-U. Rehman [1, Theorem]. [ABSTRACT FROM AUTHOR]

We consider the linear cocycle ( T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of ( T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if X is a compact connected manifold, then for a C r ( r ≥ 1) open and dense set of maps A, either ( T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of ( T, A) vanish for the generic measurable T. For the continuous case, we obtain that if X is of dimension greater than 2, then for a C r ( r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either ( T, A) is uniformly hyperbolic or the Lyapunov exponents of ( T, A) vanish. [ABSTRACT FROM AUTHOR]

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W t − I ∈ S 2 (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations. [ABSTRACT FROM AUTHOR]

Interval exchange transformations (IETs) are piecewise isometries of the interval, obtained permuting a certain number of subintervals. We give a condition on IETs in the special subclass of IETs with periodic Rauzy-Veech cocycle which guarantees weak mixing, i.e. the continuity of the spectrum. The proof involves the study of the associated spectral measures. The condition can be checked explicitly by computing a certain Galois group of a field related to the Ravzy-Veech cocycle. Explicit examples of weakly mixing IETs are constructed in the Appendix. [ABSTRACT FROM AUTHOR]

Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τrT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τrS(α,β) by a similar method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalization of Vámos's proposition which asserts that finitely generated artinian and distributive modules are cyclic. Also, we show its duality as an answer to the problem which is shown by him (See Vámos, [Vámos, P. (1978). Finitely generated artinian and distributive modules are cyclic. Bull. London Math. Soc. 10(3):287–288]). [ABSTRACT FROM AUTHOR]

Let φ a minimal affine Z[supp]-action on the torus T[supq], p ≥ 2 and q ≥ 1. The cohomology of φ (see definition below) depends on both the algebraic properties of the induced action on H[sup1](T[supq], Z) and the arithmetical properties of the translation cocycle. We give a Diophantine condition that characterizes those affine actions whose first cohomology group is finite dimensional. In this case it is necessarily isomorphic to R[supp]. Thus the R[supp]-action F[subφ] obtained by suspension of φ is parameter rigid, i.e., any other R[supp]-action with the same orbit foliation is smoothly conjugate to a reparametrization of F[subφ] by an automorphism of R[supp]. [ABSTRACT FROM AUTHOR]

For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators [$\{\backslash cal\; F\}$,ρ(x)] for some self-adjoint operator $\{\backslash cal\; F\}$ and some *-representation ρ of the underlying *-algebra. [ABSTRACT FROM AUTHOR]

We show that the sum of the first κ Lyapunov exponents of linear cocycles is an upper semicontinuous function in the L[SUPp] topologies, for any 1 ≥ p ≥ ∞ and κ. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the L[SUPp]-generic cocycle, p < ∞, are all equal. [ABSTRACT FROM AUTHOR]