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Journal articles on the topic 'Cocycles'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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1

SADOVSKAYA, VICTORIA. "Cohomology of fiber bunched cocycles over hyperbolic systems." Ergodic Theory and Dynamical Systems 35, no. 8 (August 4, 2014): 2669–88. http://dx.doi.org/10.1017/etds.2014.43.

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We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

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2

DAMJANOVIĆ, DANIJELA, and DISHENG XU. "Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions." Ergodic Theory and Dynamical Systems 40, no. 1 (April 5, 2018): 117–41. http://dx.doi.org/10.1017/etds.2018.22.

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We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

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3

FARMER, D. G., and K. J. HORADAM. "A POLYNOMIAL APPROACH TO COCYCLES OVER ELEMENTARY ABELIAN GROUPS." Journal of the Australian Mathematical Society 85, no. 2 (October 2008): 177–90. http://dx.doi.org/10.1017/s1446788708000876.

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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.

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4

Manuilov, Vladimir, and Chao You. "Vector bundles from generalized pairs of cocycles." International Journal of Mathematics 25, no. 06 (June 2014): 1450061. http://dx.doi.org/10.1142/s0129167x1450061x.

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It is interesting to know, how far we can generalize the notion of a group-valued cocycle keeping the property to determine a bundle. We find a generalization for pairs of cocycles and show how these generalized pairs of cocycles can still determine vector bundles.

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5

LINDSAY, J. MARTIN, and STEPHEN J. WILLS. "Quantum stochastic operator cocycles via associated semigroups." Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 3 (May 2007): 535–56. http://dx.doi.org/10.1017/s0305004106009923.

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AbstractA recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

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6

KARAKHANYAN, D. R., R. P. MANVELYAN, and R. L. MKRTCHYAN. "TRACE ANOMALIES AND COCYCLES OF WEYL AND DIFFEOMORPHISMS GROUPS." Modern Physics Letters A 11, no. 05 (February 20, 1996): 409–21. http://dx.doi.org/10.1142/s021773239600045x.

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The general structure of trace anomaly, suggested recently by Deser and Schwimmer is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in d=2k dimensions is considered, and explicit answers for d=4, 6 are obtained. In particular, it is shown that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field σ, i.e. the form, similar to the two-dimensional Liouville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order d and zero weight. We also give the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group.

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7

Mortier, Arnaud. "Finite-type 1-cocycles of knots given by Polyak–Viro Formulas." Journal of Knot Theory and Its Ramifications 24, no. 10 (September 2015): 1540004. http://dx.doi.org/10.1142/s0218216515400040.

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We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak–Viro's formulas for finite-type knot invariants. We conjecture that these 1-cocycles represent finite-type cohomology classes in the sense of Vassiliev. An example of degree 3 is studied, and shown to coincide over ℤ2 with the Teiblum–Turchin cocycle [Formula: see text].

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8

Clark, W. Edwin, Larry A. Dunning, and Masahico Saito. "Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles." Journal of Knot Theory and Its Ramifications 26, no. 07 (February 24, 2017): 1750035. http://dx.doi.org/10.1142/s0218216517500353.

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We explore a knot invariant derived from colorings of corresponding [Formula: see text]-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle [Formula: see text]-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding [Formula: see text]-cocycles. This permits the construction of many [Formula: see text]-cocycle invariants without exhibiting explicit [Formula: see text]-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann’s knot coloring polynomial. Computations using this technique show that the [Formula: see text]-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.

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9

BOCKER-NETO, CARLOS, and MARCELO VIANA. "Continuity of Lyapunov exponents for random two-dimensional matrices." Ergodic Theory and Dynamical Systems 37, no. 5 (March 8, 2016): 1413–42. http://dx.doi.org/10.1017/etds.2015.116.

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10

KATOK, ANATOLE, VIOREL NIŢICĂ, and ANDREI TÖRÖK. "Non-abelian cohomology of abelian Anosov actions." Ergodic Theory and Dynamical Systems 20, no. 1 (February 2000): 259–88. http://dx.doi.org/10.1017/s0143385700000122.

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We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

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11

SADEL, CHRISTIAN, and DISHENG XU. "Singular analytic linear cocycles with negative infinite Lyapunov exponents." Ergodic Theory and Dynamical Systems 39, no. 4 (August 17, 2017): 1082–98. http://dx.doi.org/10.1017/etds.2017.52.

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We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the $k$th Lyapunov exponent is finite and the $(k+1)$st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.

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12

Backes, Lucas. "Cohomology of fiber-bunched twisted cocycles over hyperbolic systems." Proceedings of the Edinburgh Mathematical Society 63, no. 3 (July 21, 2020): 844–60. http://dx.doi.org/10.1017/s0013091520000206.

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AbstractA twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

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13

Chemnitz, Robin, Maximilian Engel, and Péter Koltai. "Continuous-time extensions of discrete-time cocycles." Proceedings of the American Mathematical Society, Series B 11, no. 3 (March 5, 2024): 23–35. http://dx.doi.org/10.1090/bproc/209.

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We consider linear cocycles taking values in S L d ( R ) \mathrm {SL}_d(\mathbb {R}) driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in S L 2 ( R ) \mathrm {SL}_{2}(\mathbb {R}) over a uniquely ergodic driving.

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14

Moraga, Alexis, and Mario Ponce. "Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Lǐsic case*." Nonlinearity 35, no. 4 (February 16, 2022): 1634–51. http://dx.doi.org/10.1088/1361-6544/ac4f34.

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Abstract We consider the reducibility problem of cocycles by isometries of Gromov hyperbolic metric spaces in the Lǐsic setting. We show that provided that the boundary cocycle (that acts on a compact space) is reducible in a suitable Hölder class, then the original cocycle by isometries (that acts on an unbounded space) is also reducible.

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15

Hassanzadeh, Mohammad. "On Cyclic Cohomology of ×-Hopf algebras." Journal of K-Theory 13, no. 1 (January 2, 2014): 147–70. http://dx.doi.org/10.1017/is013011021jkt246.

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AbstractIn this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

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16

BOCHI, JAIRO, and ANDRÉS NAVAS. "A geometric path from zero Lyapunov exponents to rotation cocycles." Ergodic Theory and Dynamical Systems 35, no. 2 (August 20, 2013): 374–402. http://dx.doi.org/10.1017/etds.2013.58.

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AbstractWe consider cocycles of isometries on spaces of non-positive curvature $H$. We show that the supremum of the drift over all invariant ergodic probability measures equals the infimum of the displacements of continuous sections under the cocycle dynamics. In particular, if a cocycle has uniform sublinear drift, then there are almost invariant sections, that is, sections that move arbitrarily little under the cocycle dynamics. If, in addition, $H$ is a symmetric space, then we show that almost invariant sections can be made invariant by perturbing the cocycle.

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17

Laureano, Rosário D. "Livschitz Theorem in Suspension Flows and Markov Systems: Approach in Cohomology of Systems." Symmetry 12, no. 3 (February 27, 2020): 338. http://dx.doi.org/10.3390/sym12030338.

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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.

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18

ARNOLD, LUDWIG, and NGUYEN DINH CONG. "On the simplicity of the Lyapunov spectrum of products of random matrices." Ergodic Theory and Dynamical Systems 17, no. 5 (October 1997): 1005–25. http://dx.doi.org/10.1017/s0143385797086355.

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Assuming that the underlying probability space is non-atomic, we prove that products of random matrices (linear cocycles) with simple Lyapunov spectrum form an $L^p$-dense set ($1 \leq p < \infty$) in the space of all cocycles satisfying the integrability conditions of the multiplicative ergodic theorem. However, the linear cocycles with one-point spectrum are also $L^p$-dense. Further, in any $L^\infty$-neighborhood of an orthogonal cocycle there is a diagonalizable cocycle.For products of independent identically distributed random matrices (with distribution $\mu$), simplicity of the Lyapunov spectrum holds on a set of $\mu$'s which is open and dense in both the topology of total variation and the topology of weak convergence, hence is generic in both topologies. For products of matrices which form a Markov chain, the spectrum is simple on a set of transition functions dense in the topology of weak convergence.

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19

Muhly, Paul S., and Baruch Solel. "Representations of triangular subalgebras of groupoid C*-algebras." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 61, no. 3 (December 1996): 289–321. http://dx.doi.org/10.1017/s1446788700000392.

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AbstractWe investigate the invariant subspace structure of subalgebras of groupoid C*-algebras that are determined by automorphism groups implemented by cocycles on the groupoids. The invariant subspace structure is intimately tied to the asymptotic behavior of the cocycle.

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20

AMEUR, KHEIRA, and MASAHICO SAITO. "POLYNOMIAL COCYCLES OF ALEXANDER QUANDLES AND APPLICATIONS." Journal of Knot Theory and Its Ramifications 18, no. 02 (February 2009): 151–65. http://dx.doi.org/10.1142/s0218216509006938.

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Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.

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21

IWAKIRI, MASAHIDE. "CALCULATION OF DIHEDRAL QUANDLE COCYCLE INVARIANTS OF TWIST SPUN 2-BRIDGE KNOTS." Journal of Knot Theory and Its Ramifications 14, no. 02 (March 2005): 217–29. http://dx.doi.org/10.1142/s0218216505003798.

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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.

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22

Carter, J. Scott, Mohamed Elhamdadi, Marina Appiou Nikiforou, and Masahico Saito. "Extensions of Quandles and Cocycle Knot Invariants." Journal of Knot Theory and Its Ramifications 12, no. 06 (September 2003): 725–38. http://dx.doi.org/10.1142/s0218216503002718.

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Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

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23

FROYLAND, GARY, SIMON LLOYD, and ANTHONY QUAS. "Coherent structures and isolated spectrum for Perron–Frobenius cocycles." Ergodic Theory and Dynamical Systems 30, no. 3 (September 4, 2009): 729–56. http://dx.doi.org/10.1017/s0143385709000339.

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AbstractWe present an analysis of one-dimensional models of dynamical systems that possess ‘coherent structures’: global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalize the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.

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24

TSUTSUI, IZUMI. "COCYCLES IN TOPOLOGICAL YANG-MILLS THEORIES." Modern Physics Letters A 03, no. 01 (January 1988): 11–17. http://dx.doi.org/10.1142/s0217732388000039.

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We show that the cocycles arise in the Yang-Mills theories with topological terms added. They are found in the representation of the translation group in the configuration space or gauge orbit space, and are shown to be directly related to the U(1) functional connection in these spaces. In 3+1 dimensions, if we adopt a vector representation (i.e. vanishing 2-cocycle), we find the θ-parameter in the Pontryagin term to be 2π×integers. While, in 2+1 dimensions, insisting that finite translations be associative (i.e. vanishing 3-cocycle) leads to the known quantization of the topological mass. Some related aspects with anomalies are also discussed.

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25

ASAMI, SOICHIRO. "THE INTEGRALITY OF DIHEDRAL QUANDLE COCYCLE INVARIANTS." Journal of Knot Theory and Its Ramifications 14, no. 07 (November 2005): 953–62. http://dx.doi.org/10.1142/s0218216505004172.

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We calculate the dihedral quandle cocycle invariants of twist-spins of alternating odd pretzel knots. The calculation leads us to the conclusion that there exist non-ribbon 2-knots which admit a non-trivial coloring by the dihedral quandle Rp and all of whose cocycle invariants derived from ℤp-valued 3-cocycles on Rp take value in ℤ ⊂ ℤ[ℤp] for any odd prime integer p.

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26

DRAGIČEVIĆ, DAVOR, and GARY FROYLAND. "Hölder continuity of Oseledets splittings for semi-invertible operator cocycles." Ergodic Theory and Dynamical Systems 38, no. 3 (September 9, 2016): 961–81. http://dx.doi.org/10.1017/etds.2016.55.

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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.

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27

MADDEN, K. M., N. G. MARKLEY, and M. SEARS. "The structural stability of topological cocycles." Ergodic Theory and Dynamical Systems 19, no. 5 (October 1999): 1309–24. http://dx.doi.org/10.1017/s0143385799141750.

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Cocycles of $Z^m$ actions on compact metric spaces can be used to construct $R^m$ actions or flows, called suspension flows. A suspension provides a higher-dimensional analog to the familiar flow under a function and we look to this construction as a way of generating interesting $R^m$ flows. Even more importantly, an $R^m$ flow with a free dense orbit has an almost one-to-one extension which is a suspension [6] and thus suspensions can be used to model general $R^m$ flows. In this paper we examine the sensitivity of the suspension construction to small perturbations in the cocycle. Theorem 4.7 establishes the fact that two cocycles that are sufficiently close yield suspensions that are isomorphic up to a time change.

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28

INOUE, AYUMU. "QUASI-TRIVIALITY OF QUANDLES FOR LINK-HOMOTOPY." Journal of Knot Theory and Its Ramifications 22, no. 06 (May 2013): 1350026. http://dx.doi.org/10.1142/s0218216513500260.

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We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.

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29

SCHMIDT, KLAUS. "Tilings, fundamental cocycles and fundamental groups of symbolic ${\Bbb Z}^{d}$-actions." Ergodic Theory and Dynamical Systems 18, no. 6 (December 1998): 1473–525. http://dx.doi.org/10.1017/s0143385798118060.

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We prove that certain topologically mixing two-dimensional shifts of finite type have a ‘fundamental’ $1$-cocycle with the property that every continuous $1$-cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of Conway, Lagarias and Thurston, and they determine the projective fundamental groups of the shift spaces introduced by Geller and Propp.

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30

RAEBURN, IAIN. "Deformations of Fell bundles and twisted graph algebras." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 3 (May 24, 2016): 535–58. http://dx.doi.org/10.1017/s0305004116000359.

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AbstractWe consider Fell bundles over discrete groups, and theC*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as theC*-algebra of a Fell bundle, and there are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving that the deformations associated to different cocycles can be assembled as the fibres of aC*-bundle.

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31

ARBIETO, ALEXANDER, and JAIRO BOCHI. "Lp-GENERIC COCYCLES HAVE ONE-POINT LYAPUNOV SPECTRUM." Stochastics and Dynamics 03, no. 01 (March 2003): 73–81. http://dx.doi.org/10.1142/s0219493703000619.

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We show that the sum of the first k Lyapunov exponents of linear cocycles is an upper semicontinuous function in the Lp topologies, for any 1 ≤ p ≤ ∞ and k. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the Lp-generic cocycle, p < ∞, are all equal.

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32

KUNO, YUSUKE. "A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group." Mathematical Proceedings of the Cambridge Philosophical Society 146, no. 1 (January 2009): 109–18. http://dx.doi.org/10.1017/s0305004108001680.

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AbstractWe present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

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33

Clark, W. Edwin, and Masahico Saito. "Algebraic properties of quandle extensions and values of cocycle knot invariants." Journal of Knot Theory and Its Ramifications 25, no. 14 (December 2016): 1650080. http://dx.doi.org/10.1142/s0218216516500802.

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Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial [Formula: see text]-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

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34

Duarte, Pedro, and Silvius Klein. "Large deviation type estimates for iterates of linear cocycles." Stochastics and Dynamics 16, no. 03 (March 8, 2016): 1660010. http://dx.doi.org/10.1142/s0219493716600108.

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We describe some methods used to derive large deviation type (LDT) estimates for quantities associated to random and quasi-periodic linear cocycles. We then explain how such LDT estimates can be used in an inductive scheme to prove continuity properties of the Lyapunov exponents as functions of the cocycle. This is a survey of recent work to appear in a research monograph.

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35

JENKINSON, OLIVER. "Smooth cocycle rigidity for expanding maps, and an application to Mostow rigidity." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 3 (May 2002): 439–52. http://dx.doi.org/10.1017/s0305004101005710.

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We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.

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36

Amosov, G. G. "On Markovian cocycle perturbations in classical and quantum probability." International Journal of Mathematics and Mathematical Sciences 2003, no. 54 (2003): 3443–67. http://dx.doi.org/10.1155/s0161171203211200.

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We introduce Markovian cocycle perturbations of the groups of transformations associated with classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian cocycle perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed into the sum of two parts. One part in the decomposition is associated with a deterministic stochastic process lying in the past of the initial process, while another part is associated with the noise isomorphic to the initial one. This construction can be considered as some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increments, the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming the Gaussian stateρto the Gaussian states equivalent toρ.

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37

FROYLAND, GARY, and OGNJEN STANCEVIC. "METASTABILITY, LYAPUNOV EXPONENTS, ESCAPE RATES, AND TOPOLOGICAL ENTROPY IN RANDOM DYNAMICAL SYSTEMS." Stochastics and Dynamics 13, no. 04 (October 7, 2013): 1350004. http://dx.doi.org/10.1142/s0219493713500044.

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We explore the concept of metastability in random dynamical systems, focusing on connections between random Perron–Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron–Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.

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Koytcheff, Robin. "Bott–Taubes–Vassiliev cohomology classes by cut-and-paste topology." International Journal of Mathematics 30, no. 10 (September 2019): 1950047. http://dx.doi.org/10.1142/s0129167x19500472.

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Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct “Vassiliev classes” in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in [Formula: see text] for [Formula: see text]. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configuration spaces. Our classes form the integer lattice among the previously discovered real cohomology classes. Thus we obtain nontrivial classes from trivalent graph cocycles. Our methods generalize to yield mod-[Formula: see text] classes out of mod-[Formula: see text] graph cocycles, which need not be reductions of classes over the integers.

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Ishii, Atsushi, Masahide Iwakiri, Seiichi Kamada, Jieon Kim, Shosaku Matsuzaki, and Kanako Oshiro. "Biquandle (co)homology and handlebody-links." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843011. http://dx.doi.org/10.1142/s0218216518430113.

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In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for [Formula: see text]-oriented handlebody-links using [Formula: see text]-cocycles. When a multiple conjugation biquandle [Formula: see text] is obtained from a biquandle [Formula: see text] using [Formula: see text]-parallel operations, we provide a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the multiple conjugation biquandle [Formula: see text] from a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the biquandle [Formula: see text] equipped with an [Formula: see text]-set [Formula: see text].

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FROYLAND, GARY, CECILIA GONZÁLEZ-TOKMAN, and RUA MURRAY. "Quenched stochastic stability for eventually expanding-on-average random interval map cocycles." Ergodic Theory and Dynamical Systems 39, no. 10 (January 25, 2018): 2769–92. http://dx.doi.org/10.1017/etds.2017.143.

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The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] 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 Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

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41

Bai, Ruipu, and Ying Li. "Tθ∗-Extensions of n-Lie Algebras." ISRN Algebra 2011 (September 11, 2011): 1–11. http://dx.doi.org/10.5402/2011/381875.

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The paper is mainly concerned with Tθ∗-extensions of n-Lie algebras. The Tθ∗-extension Lθ(L∗) of an n-Lie algebra L by a cocycle θ is defined, and a class of cocycles is constructed by means of linear mappings from an n-Lie algebra on to its dual space. Finally all Tθ∗-extensions of (n+1)-dimensional n-Lie algebras are classified, and the explicit multiplications are given.

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MELNICK, KARIN. "Non-stationary smooth geometric structures for contracting measurable cocycles." Ergodic Theory and Dynamical Systems 39, no. 2 (June 28, 2017): 392–424. http://dx.doi.org/10.1017/etds.2017.38.

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We implement a differential-geometric approach to normal forms for contracting measurable cocycles to $\operatorname{Diff}^{q}(\mathbb{R}^{n},\mathbf{0})$, $q\geq 2$. We obtain resonance polynomial normal forms for the contracting cocycle and its centralizer, via $C^{q}$ changes of coordinates. These are interpreted as non-stationary invariant differential-geometric structures. We also consider the case of contracted foliations in a manifold, and obtain $C^{q}$ homogeneous structures on leaves for an action of the group of subresonance polynomial diffeomorphisms together with translations.

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Huang, Hua-Lin, Zheyan Wan, and Yu Ye. "EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 4 (March 13, 2019): 1937–64. http://dx.doi.org/10.1017/prm.2019.15.

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AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

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WIESBROCK, HANS-WERNER. "SUPERSELECTION STRUCTURE AND LOCALIZED CONNES’ COCYCLES." Reviews in Mathematical Physics 07, no. 01 (January 1995): 133–60. http://dx.doi.org/10.1142/s0129055x95000086.

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Let ρ be a localized endomorphism of the universal algebra of observables of a chiral conformal quantum field theory on a circle, see [16, 17, 23] or Chapter 1. Then ρ transforms covariant under the Möbius group. As was pointed out by D. Guido and R. Longo, [23], the covariance transformations are implemented by [Formula: see text] where Ad ∆it are modular groups to local algebras w.r.t. the vacuum vector, ut is a Connes-Radon-Nikodym-Cocycle. Using the localization property of ρ, one gets, at least for regular nets, localization properties of the cocycles. In this work we will do some steps into the opposite direction. Given a localized Connes’ cocycle of a local algebra. We will construct a localized endomorphism on the whole net. The features of this approach are twofold. Firstly sectors of finite and infinite statistical dimensions are handled on the same footing. Secondly it is a local theory right from the beginning. Moreover, soliton-like sectors can easily be incorporated. We will sketch on the last part. The program is carried through for a special class of conformal quantum field theories, the strongly additive ones.

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Elhamdadi, Mohamed, Minghui Liu, and Sam Nelson. "Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843007. http://dx.doi.org/10.1142/s0218216518430071.

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We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

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46

Molaei, Mohammad Reza, and Tahere Nasirzadeh. "On cocycles." Journal of Interdisciplinary Mathematics 22, no. 1 (January 2, 2019): 91–99. http://dx.doi.org/10.1080/09720502.2019.1575048.

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47

Avila, Artur, and Raphaël Krikorian. "Monotonic cocycles." Inventiones mathematicae 202, no. 1 (February 3, 2015): 271–331. http://dx.doi.org/10.1007/s00222-014-0572-6.

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48

Bonora, L., P. Pasti, and M. Bregola. "Weyl cocycles." Classical and Quantum Gravity 3, no. 4 (July 1, 1986): 635–49. http://dx.doi.org/10.1088/0264-9381/3/4/018.

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Molaei, M. R. "Topological cocycles." Boletín de la Sociedad Matemática Mexicana 24, no. 1 (January 3, 2017): 257–67. http://dx.doi.org/10.1007/s40590-016-0158-y.

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LEMAŃCZYK, MARIUSZ, and FRANÇOIS PARREAU. "Lifting mixing properties by Rokhlin cocycles." Ergodic Theory and Dynamical Systems 32, no. 2 (November 8, 2011): 763–84. http://dx.doi.org/10.1017/s0143385711000666.

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AbstractWe study the problem of lifting various mixing properties from a base automorphismT∈Aut(X,ℬ,μ) to skew products of the formTφ,𝒮, where φ:X→Gis a cocycle with values in a locally compact Abelian groupG, 𝒮=(Sg)g∈Gis a measurable representation ofGinAut(Y,𝒞,ν) andTφ,𝒮acts on the product space (X×Y,ℬ⊗𝒞,μ⊗ν) byIt is also shown that wheneverTis ergodic (mildly mixing, mixing) butTφ,𝒮is not ergodic (is not mildly mixing, not mixing), then, on a non-trivial factor 𝒜⊂𝒞 of 𝒮, the corresponding Rokhlin cocyclex↦Sφ(x)∣𝒜is a coboundary (a quasi-coboundary).

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