On biunitary permutation matrices and some subfactors of index 9
Uma
Krishnan;
V.
S.
Sunder
4691-4736
Abstract: This paper is devoted to a study of the subfactors arising from vertex models constructed out of `biunitary' matrices which happen to be permutation matrices. After a discussion on the computation of the higher relative commutants of the associated subfactor (in the members of the tower of Jones' basic construction), we discuss the principal graphs of these subfactors for small sizes $N=k \leq 3$ of the vertex model. Of the 18 possibly inequivalent such biunitary matrices when $N = 3$, we compute the principal graphs completely in 15 cases, all of which turn out to be finite. In the last section, we prove that two of the three remaining cases lead to subfactors of infinite depth and discuss their principal graphs.
On the group of homotopy equivalences of a manifold
Hans
Joachim
Baues
4737-4773
Abstract: We consider the group of homotopy equivalences $\mathcal E(M)$ of a simply connected manifold $M$ which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group $\mathcal E(M)$ for special manifolds $M$, for example if $M$ is a connected sum of products $S^n\times S^m$ of spheres. In particular the group $\mathcal E(S^n\times S^n)$ is determined completely. Also the connection of $\mathcal E(M)$ with the group of isotopy classes of diffeomorphisms of $M$ is studied.
The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type
Birge
Huisgen-Zimmermann
4775-4812
Abstract: A description is given of those sequences $\mathbf {S}= (S(0),S(1),\dots ,S(l))$ of simple modules over a finite dimensional algebra for which there are only finitely many uniserial modules with consecutive composition factors $S(0),\dots , S(l)$. Necessary and sufficient conditions for an algebra to permit only a finite number of isomorphism types of uniserial modules are derived. The main tools in this investigation are the affine algebraic varieties parametrizing the uniserial modules with composition series $\mathbf {S}$.
Multiplication of natural number parameters and equations in a free semigroup
Gennady
S.
Makanin
4813-4824
Abstract: This paper deals with the problem of describing the set $M$ of all solutions of an equation over a free semigroup $S$. The standard way to do this involves the introduction of auxiliary equations containing polynomials in natural number parameters of arbitrarily high degree. Since $S$ has a solvable word problem, $M$ must be computable. However, $M$ cannot necessarily be computed from the standard description of $M$. The present paper shows that the only polynomials needed to describe $M$ are just products of one parameter by a linear combination of some other parameters. The resulting simplification of the standard description of $M$ clearly can be used to compute $M$.
Multiplicative $\eta$-quotients
Yves
Martin
4825-4856
Abstract: Let $\eta (z)$ be the Dedekind $\eta $-function. In this work we exhibit all modular forms of integral weight $f(z) = \eta (t_1z)^{r_1}\eta (t_2z)^{r_2}\dots \eta(t_sz)^{r_s}$, for positive integers $s$ and $t_j$ and arbitrary integers $r_j$, such that both $f(z)$ and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group $2 % \mathrm {Co}_1$ via a generalized McKay-Thompson series.
On extension of cocycles to normalizer elements, outer conjugacy, and related problems
Alexandre
I.
Danilenko;
Valentin
Ya.
Golodets
4857-4882
Abstract: Let $T$ be an ergodic automorphism of a Lebesgue space and $\alpha$ a cocycle of $T$ with values in an Abelian locally compact group $G$. An automorphism $\theta$ from the normalizer $N[T]$ of the full group $[T]$ is said to be compatible with $\alpha$ if there is a measurable function $\varphi : X \to G$ such that $\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)$ at a.e. $x$. The topology on the set $D(T, \alpha )$ of all automorphisms compatible with $\alpha$ is introduced in such a way that $D(T , \alpha )$ becomes a Polish group. A complete system of invariants for the $\alpha$-outer conjugacy (i.e. the conjugacy in the quotient group $D(T, \alpha )/[T])$ is found. Structure of the cocycles compatible with every element of $N[T]$ is described.
Spectral averaging, perturbation of singular spectra, and localization
J.
M.
Combes;
P.
D.
Hislop;
E.
Mourre
4883-4894
Abstract: A spectral averaging theorem is proved for one-parameter families of self-adjoint operators using the method of differential inequalities. This theorem is used to establish the absolute continuity of the averaged spectral measure with respect to Lebesgue measure. This is an important step in controlling the singular continuous spectrum of the family for almost all values of the parameter. The main application is to the problem of localization for certain families of random Schrödinger operators. Localization for a family of random Schrödinger operators is established employing these results and a multi-scale analysis.
Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions
V.
I.
Chilin;
P.
G.
Dodds;
A.
A.
Sedaev;
F.
A.
Sukochev
4895-4918
Abstract: We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the $K$-functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.
Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces
M.
Lemanczyk;
F.
Parreau;
D.
Volný
4919-4938
Abstract: Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior. Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In ``natural" cases, up to $L^1$-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials. For pseudo-homogeneous spaces admitting a Koksma's inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.
A deformation of flat conformal structures
Hiroyasu
Izeki
4939-4964
Abstract: We consider deformations of flat conformal structures from a viewpoint of connected sum decomposition of conformally flat manifolds.
The ergodic theory of discrete isometry groups on manifolds of variable negative curvature
Chengbo
Yue
4965-5005
Abstract: This paper studies the ergodic theory at infinity of an arbitrary discrete isometry group $\Gamma$ acting on any Hadamard manifold $H$ of pinched variable negative curvature. Most of the results obtained by Sullivan in the constant curvature case are generalized to the case of variable curvature. We describe connections between measures supported on the limit set of $\Gamma$, dynamics of the geodesic flow and the geometry of $M=H/ \Gamma$. We explore the relationship between the growth exponent of the group, the Hausdorff dimension of the limit set and the topological entropy of the geodesic flow. The equivalence of various descriptions of an analogue of the Hopf dichotomy is proved. As applications, we settle a question of J. Feldman and M. Ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points. At the end of this paper, we apply our results to the study of some rigidity problems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with infinite covolume will be published in subsequent papers by the author.
The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations
M.
E.
Fels
5007-5029
Abstract: A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.
Spatial chaotic structure of attractors of reaction-diffusion systems
V.
Afraimovich;
A.
Babin;
S.-N.
Chow
5031-5063
Abstract: The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.
Integral type linear functionals on ordered cones
Walter
Roth
5065-5085
Abstract: We introduce linear functionals on an ordered cone that are minimal with respect to a given subcone. Using concepts developed for Choquet theory we observe that the properties of these functionals resemble those of positive Radon measures on locally compact spaces. Other applications include monotone functionals on cones of convex sets, H-integrals on H-cones in abstract potential theory, and classical Choquet theory itself.
Lévy group action and invariant measures on $\beta\mathbb{N}$
Martin
Blümlinger
5087-5111
Abstract: For $f\in \ell ^{\infty }( \mathbb {N})$ let $Tf$ be defined by $Tf(n)=\frac {1}{n}\sum _{i=1}^{n}f(i)$. We investigate permutations $g$ of $ \mathbb {N}$, which satisfy $Tf(n)-Tf_{g}(n)\to 0$ as $n\to \infty$ with $f_{g}(n)=f(gn)$ for $f\in \ell ^{\infty }( \mathbb {N})$ (i.e. $g$ is in the Lévy group $\mathcal {G})$, or for $f$ in the subspace of Cesàro-summable sequences. Our main interest are $ \mathcal {G}$-invariant means on $\ell ^{\infty }( \mathbb {N})$ or equivalently $\mathcal {G}$-invariant probability measures on $\beta \mathbb {N}$. We show that the adjoint $T^{*}$ of $T$ maps measures supported in $\beta \mathbb {N} \setminus \mathbb {N}$ onto a weak*-dense subset of the space of $ \mathcal {G}$-invariant measures. We investigate the dynamical system $( \mathcal {G}, \beta \mathbb {N})$ and show that the support set of invariant measures on $\beta \mathbb {N}$ is the closure of the set of almost periodic points and the set of non-topologically transitive points in $\beta \mathbb {N}\setminus \mathbb {N}$. Finally we consider measures which are invariant under $T^{*}$.