Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.     

Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

 This paper describes the cutting sequences of geodesic flow on the modular surface with respect to the standard fundamental domain of . The cutting sequence for a vertical geodesic is related to a one-dimensional continued fraction expansion for θ, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain ℱ up to an isometry of the hyperbolic plane . We give conversion methods between the cutting sequence for the vertical geodesic , the MGCF expansion of θ and the additive ordinary continued fraction (ACF) expansion of θ. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of θ can be computed from the cutting sequence for the vertical geodesic θ + it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first symbols when given as input the first symbols of the ACF expansion, which takes time and space . (Received 11 August 2000; in revised form 18 April 2001)     

On convolution equations with semi-almost periodic symbols on a finite interval

This note concerns a class of Wiener-Hopf operators on a finite interval, acting between Sobolev multi-index spaces. Necessary and sufficient conditions for such an operator to be Fredholm are given, as well as a formula for the index. The argument is based on a reduction procedure of convolution operators on a finite interval to operators of the same type on the half-line.supported by the Netherlands organization for scientific research (NWO)supported in part by NSF Grant 9101143     

Construction of covering maps between orientable surfaces

Consider the following problem. Can a set of numbers be realized as boundary covering indices of a covering map between surfaces? How to realize them? A set of equivalent criteria for this problem are found, which can be checked by a finite algorithm. Furthermore, the algorithm will also construct a solution if such one exists. As an application, a well_known group of necessary conditions are shown to be not sufficient in infinitely many cases, while in most cases, numbers satisfying them are realizable.     

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.     

Realizability and automatic realizability of Galois groups of order 32

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

On interfaces between incompatible wells and elastic deformations of infinite cylinders

We consider the minimization problem for the functional where is an infinitely long cylinder. The density is polyconvex and assumed to be 0 on a set of wells and positive elsewhere. We show that the gradients of solutions with finite energy have to approach one component for and one component for , if the number of components is finite (among other conditions). Moreover, for certain pairs of distinct components we construct nontrivial minimizers within the class of solutions approaching the given components. We follow ideas developed in the variational study of heteroclinic connections for Lagrangian systems and we put special emphasis on multiplicity of such interface solutions. We discuss an application in the theory of nonlinear elasticity, where such solutions are called semi-necks. When a two-dimensional infinite hyperelastic strip is stretched along its infinite direction it may occur that for a given tensile load many homogeneous deformations are possible. In such a case we show by infimizing the energy functional the existence of configurations that tend asymptotically to two different homogeneous deformations. Received: 1 March 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Orthogonal one-factorization graphs

An orthogonal one-factorization graph (OOFG) is a graph in which the vertices are one-factorizations of some underlying graph H, and two vertices are adjacent if and only if the one-factorizations are orthogonal. An arbitrary finite graph, G, is realizable if there is an OOFG isomorphic to G. We show that every finite graph is realizable as an OOFG with underlying graph K_n for some n. We also discuss some special cases.     

Configurations of limit cycles and planar polynomial vector fields

We show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles.     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.     

Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

Strong approximation of partial sums under dependence conditions with application to dynamical systems

In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) [30] as a special case. Applications to unbounded functions of intermittent maps are given.     

The linking pairings of orientable Seifert manifolds

We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free S¹-action. Any linking pairing on a finite abelian group of odd order is realized by such a manifold. We find necessary and sufficient conditions for a pairing on an abelian 2-group to be the 2-primary component of such a linking pairing, and give simple examples which are not realizable by any Seifert fibred 3-manifold.     

复合形在欧氏空间中的实现问题(Ⅱ)

<正> 并于复合形或更一般的空间在欧氏空间中的实现问题,曾经有过下面几个重要的结果:1°Van Kampen 在1932时证明有不能在2n维欧氏空间中实现的 n 维复合形 K 存在。Van Kampen 的证明倚赖于由及的约化二重对称积 K 作出的一个不变量.作者在[2]中指出 Van Kampen 不变量只是一组不变量(?),(I_m=整数加法群 I 或法2整数群 I_2,视 m 为偶或奇而定)其中极端的一个,即Φ~(2n),而Φ~m=0为     

Complexity of Chow varieties and number of morphisms on surfaces of general type

We provide a general estimate for the number of irreducible components of a Chow variety, the variety that parametrizes algebraic cycles of given dimension and degree contained in a projective variety. The result is then applied to obtain an upper bound for the finite number of surfaces of general type that are images of a fixed surface. Received: 29 January 1998 / Revised version: 24 June 1998     

Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.     

Boundary of Anosov dynamics and evolution equations for surfaces

We show that a compact surface of genus greater than one, without focal points and a finite number of bubbles (“good” shaped regions of positive curvature) is in the closure of Anosov metrics. Compact surfaces of nonpositive curvature and genus greater than one are in the closure of Anosov metrics, by Hamilton's work about the Ricci flow. We generalize this fact to the above surfaces without focal points admitting regions of positive curvature using a “magnetic” version of the Ricci flow, the so‐called Ricci Yang‐Mills flow.