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This work is devoted to studying the relations between the asymptotic behavior for a class of hyperbolic equations with super-cubic nonlinearity and a class of heat equations, where the problem is considered in a smooth bounded three dimensional domain. Based on the extension of the Strichartz estimates to the case of bounded domain, we show the regularity of the pullback, uniform, and cocycle attractors for the non-autonomous dynamical system given by hyperbolic equation. Then we prove that all types of non-autonomous attractors converge, upper semicontiously, to the natural extension global attractor of the limit parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper we study different types of compact global attractors for non-autonomous (cocycle) dynamical systems with noncompact phase space of driving system. We establish the relations between them. [ABSTRACT FROM AUTHOR]

We show that the class B, of discrete groups which satisfy the conclusion of Popa's cocycle superrigidity theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete probability measure preserving (p.m.p.) groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups must belong to B. We also introduce a measure-conjugacy invariant called weak Pinsker entropy and show that, if G is a group in the class B, then weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G. [ABSTRACT FROM AUTHOR]

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod's original cochain definition of the Square operations. [ABSTRACT FROM AUTHOR]

Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications are numerous and well-known in these common settings. We construct P algebra analogues to a series of results from W. R. Scott's Group Theory, which gives an explicit theory of factor systems for the group case. Here P ranges over Leibniz, Zinbiel, diassociative, and dendriform algebras, which we dub "the algebras of Loday," as well as over Lie, associative, and commutative algebras. Fixing a pair of P algebras, we develop a correspondence between factor systems and extensions. This correspondence is strengthened by the fact that equivalence classes of factor systems correspond to those of extensions. Under this correspondence, central extensions give rise to 2-cocycles while split extensions give rise to (nonabelian) 2-coboundaries. [ABSTRACT FROM AUTHOR]

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL (2 , ℝ) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles. [ABSTRACT FROM AUTHOR]

We present a new, general approach to gauge theory on principal G-spectral triples, where G is a compact connected Lie group. We introduce a notion of vertical Riemannian geometry for G- C ∗ -algebras and prove that the resulting noncommutative orbitwise family of Kostant's cubic Dirac operators defines a natural unbounded K K G -cycle in the case of a principal G-action. Then, we introduce a notion of principal G-spectral triple and prove, in particular, that any such spectral triple admits a canonical factorisation in unbounded K K G -theory with respect to such a cycle: up to a remainder, the total geometry is the twisting of the basic geometry by a noncommutative superconnection encoding the vertical geometry and underlying principal connection. Using these notions, we formulate an approach to gauge theory that explicitly generalises the classical case up to a groupoid cocycle and is compatible in general with this factorisation; in the unital case, it correctly yields a real affine space of noncommutative principal connections with affine gauge action. Our definitions cover all locally compact classical principal G-bundles and are compatible with θ -deformation; in particular, they cover the θ -deformed quaternionic Hopf fibration C ∞ (S θ 7) ↩ C ∞ (S θ 4) as a noncommutative principal SU (2) -bundle. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions in the spirit of Rolewicz under which all Lyapunov exponents of a given linear cocycle are either positive or negative. As a consequence, we formulate new conditions for the existence of the so-called tempered exponential dichotomies. We consider cocycles over both maps and flows. [ABSTRACT FROM AUTHOR]

We prove a functorial correspondence between a category of logarithmic sl 2 -connections on a curve X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover. The proof is by constructing a pair of inverse functors π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor π ∗ . [ABSTRACT FROM AUTHOR]

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles. [ABSTRACT FROM AUTHOR]

We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach. [ABSTRACT FROM AUTHOR]

We consider involutive, non-degenerate, finite set-theoretic solutions of the Yang–Baxter equation (YBE). Such solutions can be always obtained using certain algebraic structures that generalize nilpotent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set-theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the YBE constructed from the set-theoretical ones. [ABSTRACT FROM AUTHOR]

We show that the Quot scheme QLn=QuotA3(IL,n) parameterising length n quotients of the ideal sheaf of a line in A3 is a global critical locus, and calculate the resulting motivic partition function (varying n), in the ring of relative motives over the configuration space of points in A3. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of n points of the ideal sheaf IC⊂OY, where C⊂Y is a smooth curve embedded in a smooth 3‐fold Y, and we compute the associated motivic partition function. The result fits into a motivic wall‐crossing type formula, refining the relation between Behrend's virtual Euler characteristic of QuotY(IC,n) and of the symmetric product SymnC. Our 'relative' analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map QLn→Symn(A3), and connections with cohomological Hall algebra representations. [ABSTRACT FROM AUTHOR]

This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmüller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra. [ABSTRACT FROM AUTHOR]

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Toward the complete classification of poly- |${\mathbb{Z}}$| group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly- |${\mathbb{Z}}$| groups of Hirsch length not greater than three on unital Kirchberg algebras up to |$KK$| -trivial cocycle conjugacy. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we shall generalize the concept of equivariance in statistics to "weak equivariance". Then, we summarize the properties of weakly equivariant estimators and their applications in statistics. At first we characterize the class of all weakly equivariant estimators. Then, we shall consider the concept of cocycles and isovariance, and so we find their connection with weakly equivariant functions. It is natural to restrict attention to the class of weakly equivariant estimator to find minimum risk weakly equivariant estimators. If the group acts in two different ways, we shall find a relation between the minimum risk equivariant and minimum risk weakly equivariant estimator under the old and new group actions. Also we shall introduce a necessary and sufficient condition for the invariance of the loss function under the new action. [ABSTRACT FROM AUTHOR]

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$ -absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$ -absorption theorem for these groups. [ABSTRACT FROM AUTHOR]

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions. [ABSTRACT FROM AUTHOR]

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Rüssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition. [ABSTRACT FROM AUTHOR]

We prove that given a pseudo- z-embedding two Knapp-Stein R-groups are isomorphic and their 2-cocycles are identical. [ABSTRACT FROM AUTHOR]

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of Cr, 0 ≤ r ≤ ∞ (or analytic) κ + 1-tuples of quasi-periodic cocycles taking values in SL2(R) such that the random product of them has positive Lyapunov exponent contains a C0 open and Cr dense subset which is formed by C0 continuity point of the Lyapunov exponent. For κ + 1-tuples of quasi-periodic cocycles taking values in GLd(R) for d > 2, we prove that if one of them is diagonal, then there exists a Cr dense set of such κ + 1-tuples which have simple Lyapunov spectrum and are C0 continuity point of the Lyapunov exponent. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of uniform pullback asymptotic compactness of a non-autonomous random dynamical system, and prove the existence of random uniform and cocycle attractor with autonomous attraction universes for non-autonomous stochastic FitzHugh–Nagumo system with multiplicative noise defined on whole space. Some properties of random uniform and cocycle attractor are shown. The method of tail estimates plays an important role. [ABSTRACT FROM AUTHOR]

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive n -ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing 2 -cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of 2-cocycles on monoidal Hom–Hopf algebras, study their properties, and extend neat lazy 2-cocycles to a Radford (m , n) -biproduct monoidal Hom–Hopf algebra. [ABSTRACT FROM AUTHOR]

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schrödinger cocycles with the C2 cos-type potentials. In particular, the Lyapunov exponent is log-Hölder continuous at each Diophantine frequency. [ABSTRACT FROM AUTHOR]

We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al.(Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zero-memory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory. [ABSTRACT FROM AUTHOR]

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of SL 2 (Z [ 1 / p ]) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which SL 2 (Z [ 1 / p ]) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture. [ABSTRACT FROM AUTHOR]

The classical Multiplicative Ergodic Theorem of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution. [ABSTRACT FROM AUTHOR]

Involving a symmetric Hochschild 1-cocycle condition, we equip the space of decorated planar rooted forests with a coproduct which turns the space into a dendriform-Nijenhuis bialgebra. We combine dendriform-Nijenhuis bialgebras with operated algebras and introduce the notation of an Ω-operated DN-bialgebra. Applying the universal property of the underlying operated algebras, we construct free objects in the category of Ω-cocycle DN-bialgebras. We introduce the notation of a DN-associative Yang-Baxter equation (AYBE) and show that the dendriform-Nijenhuis bialgebra offers an algebraic framework of the DN-associative Yang-Baxter equation. We construct a Leroux's TD operator from a solution of the DN-AYBE. We also give two different ways to derive Lie algebras from quasitriangular DN-bialgebras. Finally, we classify the solutions of the DN-AYBE in the unitary algebras of dimensions two and three over the field of complex numbers. [ABSTRACT FROM AUTHOR]

We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target. [ABSTRACT FROM AUTHOR]

A coclique extension of a graph H is a graph G obtained from H by replacing each vertex of H by a coclique, where vertices of G coming from different vertices of H are adjacent if and only if the original vertices are adjacent in H. Let M n (H) be the set of graphs with order n , which are the coclique extensions of H. In this paper, we discuss the minimum spectral radius in M n (P k) and the maximum spectral radius in M n (C k) , where P k and C k are the path of order k and the cycle of order k , respectively. Then we disprove a conjecture on the minimum spectral radius in M n (P k) and confirm a conjecture on the maximum spectral radius in M n (C k) , which are given by Monsalve and Rada (2021) [4]. [ABSTRACT FROM AUTHOR]

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that the base is compact under mild additional assumptions. Moreover, we give a few applications of the equivalence of the two notions of hyperbolicity. In particular, we consider the robustness and the admissibility of the equation, and we obtain stable and unstable invariant manifolds. [ABSTRACT FROM AUTHOR]

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite‐dimensional partial cohomological field theory given by exp(μ2Θ), where μ is a parameter and Θ is Hain's theta class, appearing in Hain's formula for the DR cycle on the moduli space of curves of compact type. This infinite rank DR hierarchy can be seen as a rank 1 integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the Korteweg‐de‐Vries (KdV) hierarchy on a noncommutative Moyal torus. [ABSTRACT FROM AUTHOR]

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat Proj -periodic module is projective, any fp-injective Inj -periodic module is injective, and any Cot -periodic module is cotorsion. It is also known that any pure PProj -periodic module is pure-projective and any pure PInj -periodic module is pure-injective. Generalizing a result of Šaroch and Št'ovíček, we show that every FpProj -periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every FpProj -periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure PProj -periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective. [ABSTRACT FROM AUTHOR]

We study the norm potentials of Hölder continuous GL2(R)-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant GL2(R)-cocycles as well as fiber-bunched GL2(R)-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states. [ABSTRACT FROM AUTHOR]

We consider cocycles obtained by composing sequences of transfer operators with positive weights, associated with uniformly expanding maps (possibly having countably many branches) and depending upon parameters. Assuming C k regularity with respect to coordinates and parameters, we show that when the sequence is picked within a certain uniform family the top characteristic exponent and generator of top Oseledets space of the cocycle are C k - 1 in parameters. As applications, we obtain a linear response formula for the equivariant measure associated with random products of uniformly expanding maps, and we study the regularity of the Hausdorff dimension of a repeller associated with random compositions of one-dimensional cookie-cutters. [ABSTRACT FROM AUTHOR]

We will show a centrally free action of an amenable rigid C ∗ -tensor category on a properly infinite von Neumann algebra has the Rohlin property. Our main result is the classification of centrally free cocycle actions of an amenable rigid C ∗ -tensor category up to approximate inner conjugacy on properly infinite von Neumann algebras. This is regarded as a generalization of classification of amenable discrete groups due to A. Connes, V. Jones and A. Ocneanu. We have the following two applications: a classification of centrally free actions of amenable discrete quantum groups of Kac type on von Neumann algebras and another proof of S. Popa's celebrated classification result of amenable subfactors. As another application of the Rohlin property, we will prove the fullness of the crossed product of a full factor by a minimal action of a compact group. [ABSTRACT FROM AUTHOR]

Quantum stochastic cocycles provide a basic model for time-homogeneous Markovian evolutions in a quantum setting, and a direct counterpart in continuous time to quantum random walks, in both the Schrödinger and Heisenberg pictures. This paper is a sequel to one in which correspondences were established between classes of quantum stochastic cocycle on an operator space or C ∗ -algebra, and classes of Schur-action 'global' semigroup on associated matrix spaces over the operator space. In this paper we investigate the stochastic generation of cocycles via the generation of their corresponding global semigroups, with the primary purpose of strengthening the scope of applicability of semigroup theory to the analysis and construction of quantum stochastic cocycles. An explicit description is given of the affine relationship between the stochastic generator of a completely bounded cocycle and the generator of any one of its associated global semigroups. Using this, the structure of the stochastic generator of a completely positive quasicontractive quantum stochastic cocycle on a C ∗ -algebra whose expectation semigroup is norm continuous is derived, giving a comprehensive stochastic generalisation of the Christensen–Evans extension of the GKS&L theorem of Gorini, Kossakowski and Sudarshan, and Lindblad. The transformation also provides a new existence theorem for cocycles with unbounded structure map as stochastic generator. The latter is applied to a model of interacting particles known as the quantum exclusion Markov process, in particular on integer lattices in dimensions one and two. [ABSTRACT FROM AUTHOR]

We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant and are not determined by the Jablan polynomial. [ABSTRACT FROM AUTHOR]

Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron–Frobenius operators associated to the maps. Of particular interest is the second-largest Lyapunov exponent for the cocycle of operators, λ2, which can tell us about mixing rates and decay of correlations in the system. We prove a generalized Perron–Frobenius theorem for cocycles of bounded linear operators on Banach spaces that preserve and occasionally contract a cone; this theorem shows that the top Oseledets space for the cocycle is one-dimensional, and there is a lower bound for the gap between the largest Lyapunov exponents λ1 and λ2 (that is, an upper bound for λ2 which is strictly less than λ1) explicitly in terms of quantities related to cone contraction. We then apply this theorem to the case of cocycles of Perron–Frobenius operators arising from a parametrized family of maps to obtain an upper bound on λ2; to the best of our knowledge, this work is the first time λ2 has been upper-bounded for a family of maps. In doing so, we utilize a new balanced Lasota–Yorke inequality. We also examine random perturbations of a fixed map within the family with two invariant densities and show that as the perturbation is scaled back down to the unperturbed map, λ2 is at least asymptotically linear in the scale parameter. Our estimates are sharp, in the sense that there is a sequence of scaled perturbations of the fixed map that are all Markov, such that λ2 is asymptotic to −2 times the scale parameter. [ABSTRACT FROM AUTHOR]

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair U(p,q) × U(1,1) to construct two different kinds of Green forms for codimension q-cycles in Shimura varieties associated to unitary groups. We establish an adjointness result between our singular theta lift and the Kudla-Millson lift. Further, we compare the two Greens forms and obtain modularity for the generating function of the difference of the two Green forms. Finally, we show that the Green forms obtained by the singular theta lift satisfy an eigenvalue equation for the Laplace operator and conclude that our Green forms coincide with the ones constructed by Oda and Tsuzuki by different means. [ABSTRACT FROM AUTHOR]

Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative (v, w, v, v/w)-difference sets. Moreover, some structural properties of these codes are studied and examples are provided. Some of the propelinear codes constructed for the examples perform better than any comparable linear code. [ABSTRACT FROM AUTHOR]

We study flows on C ∗ -algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C ∗ -algebras satisfying certain technical properties, which hold for many C ∗ -algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, O ∞ -absorbing C ∗ -algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable KK-contractible C ∗ -algebras: Two Rokhlin flows on such a C ∗ -algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate. [ABSTRACT FROM AUTHOR]

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved. [ABSTRACT FROM AUTHOR]

By generalizing Ledrappier's criterion [Mesures d'èquilibre d'entropie complètement positive, in Systèmes dynamiques II – Varsovie, number 50 in Astérisque, Société mathématique de France, 1977, pp. 251–272] for the K-property of equilibrium states, we extend the criterion to subadditive potentials. In particular, supposing that the unique equilibrium state for a subadditive potential with quasi-multiplicativity and bounded distortion is totally ergodic, we show that it has the K-property. We apply this result to subadditive potentials arising from certain classes of matrix cocycles; for the norm potentials of irreducible locally constant cocycles and the singular value potentials of typical cocycles, we show that their unique equilibrium states have the K-property. This partly generalizes the work of Morris [Ergodic properties of matrix equilibrium states, Ergodic Theory Dyn. Syst. 38(6), 2018, pp. 2295-2320] on irreducible locally constant cocycles and their subadditive equilibrium states. [ABSTRACT FROM AUTHOR]

The homotopy category of the bordism category hBordd has as objects closed oriented (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of hBordd for d=1, exhibiting it as a circle bundle over CP∞-1. As part of our proof we construct a quotient Bordred1 of the cobordism category where circles are deleted. We show that this category has classifying space Ω∞-2CP∞-1 and moreover that, if one equips these bordisms with a map to a simply connected space X, the resulting Bordred1(X) can be thought of as a cobordism model for the topological cyclic homology TC(S[ΩX]). In the second part of the paper we construct an infinite loop space map B(hBordred1)→Q(Σ²CP∞+) in this model and use it to derive combinatorial formulas for rational cocycles on Bordred1 representing the Miller-Morita-Mumford classes κiH2i+2((B(hC1);Q). [ABSTRACT FROM AUTHOR]