Finite state automata: A geometric approach
Benjamin
Steinberg
3409-3464
Abstract: Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was non-trivial), is whether it is true, for a pseudovariety of groups $\mathbf{H}$, that a ${\mathcal J}$-trivial co-extension of a group in $\mathbf{H}$ must divide a semidirect product of a ${\mathcal J}$-trivial monoid and a group in $\mathbf{H}$. We show the answer is affirmative if $\mathbf{H}$ is closed under extension, and may be negative otherwise.
Group cohomology and gauge equivalence of some twisted quantum doubles
Geoffrey
Mason;
Siu-Hung
Ng
3465-3509
Abstract: We study the module category associated to the quantum double of a finite abelian group $G$ twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of $G$. Roughly speaking, this reveals that the data provided by such module categories is equivalent to (among other things) a finite quadratic space equipped with a metabolizer, and also a pair of rational lattices $L\subseteq M$ with $L$ self-dual and integral.
The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$
Cai
Heng
Li
3511-3529
Abstract: A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text{M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text{PSp}(4,p)$ with $p$ prime and $p\equiv\pm1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.
Jordan curves in the level sets of additive Brownian motion
Robert
C.
Dalang;
T.
Mountford
3531-3545
Abstract: This paper studies the topological and connectivity properties of the level sets of additive Brownian motion. More precisely, for each excursion set of this process from a fixed level, we give an explicit construction of a closed Jordan curve contained in the boundary of this excursion set, and in particular, in the level set of this process.
Markov chains in random environments and random iterated function systems
Örjan
Stenflo
3547-3562
Abstract: We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a ``pure type'' characterization of the limiting ``randomly invariant'' probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.
Loop-erased walks and total positivity
Sergey
Fomin
3563-3583
Abstract: We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
Hausdorff convergence and universal covers
Christina
Sormani;
Guofang
Wei
3585-3602
Abstract: We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$ sufficiently large, the fundamental group of $M_i$ has a surjective homeomorphism onto the group of deck transforms of $Y$. Finally, in the non-collapsed case where the $M_i$ have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the $M_i$ are only assumed to be compact length spaces with a uniform upper bound on diameter.
Uniform perfectness of the limit sets of Kleinian groups
Toshiyuki
Sugawa
3603-3615
Abstract: In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.
Maximal degree subsheaves of torsion free sheaves on singular projective curves
E.
Ballico
3617-3627
Abstract: Fix integers $r,k,g$ with $r>k>0$ and $g\ge 2$. Let $X$ be an integral projective curve with $g:=p_a(X)$ and $E$ a rank $r$ torsion free sheaf on $X$which is a flat limit of a family of locally free sheaves on $X$. Here we prove the existence of a rank $k$ subsheaf $A$ of $E$ such that $r(\deg(A))\ge k(\deg (E))-k(r-k)g$. We show that for every $g\ge 9$ there is an integral projective curve $X,X$ not Gorenstein, and a rank 2 torsion free sheaf $E$ on $X$ with no rank 1 subsheaf $A$ with $2(\deg (A))\ge \deg(E)-g$. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
Orthogonal polynomial eigenfunctions of second-order partial differerential equations
K.
H.
Kwon;
J.
K.
Lee;
L.
L.
Littlejohn
3629-3647
Abstract: In this paper, we show that for several second-order partial differential equations \begin{align*}L[u]&=A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy}+D(x,y)u_{x}+E(x,y)u_{y} &=\lambda_{n}u \end{align*} which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.
Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations
H.
A.
Biagioni;
F.
Linares
3649-3659
Abstract: Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in $H^s(\mathbb R)$, $s<1/2$. This result implies that best result concerning local well-posedness for the IVP is in $H^s(\mathbb R),\, s\ge1/2$. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
Siegel discs, Herman rings and the Arnold family
Lukas
Geyer
3661-3683
Abstract: We show that the rotation number of an analytically linearizable element of the Arnold family $f_{a,b}(x)=x+a+b\sin(2\pi x)\pmod 1$, $a,b\in{\mathbb R}$, $0<b<1/(2\pi)$, satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of $P_{\lambda,d}(z)=\lambda z(1+z/d)^d$ and $E_\lambda(z)=\lambda z e^z$, $\lambda=e^{2\pi i\alpha}$, at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of $P_{\lambda,d}$ and $E_\lambda$ are quasidiscs with a critical point on the boundary if the rotation number is of constant type.
The structure of the Brauer group and crossed products of $C_0(X)$-linear group actions on $C_0(X,\mathcal K)$
Siegfried
Echterhoff;
Ryszard
Nest
3685-3712
Abstract: For a second countable locally compact group $G$ and a second countable locally compact space $X$let $\operatorname{Br}_G(X)$ denote the equivariant Brauer group (for the trivial $G$-space $X$) consisting of all Morita equivalence classes of spectrum fixing actions of $G$ on continuous-trace $C^*$-algebras $A$ with spectrum $\widehat{A}=X$. Extending recent results of several authors, we give a complete description of $\operatorname{Br}_G(X)$ in terms of group cohomology of $G$ and Cech cohomology of $X$. Moreover, if $G$ has a splitting group $H$ in the sense of Calvin Moore, we give a complete description of the $C_0(X)$-bundle structure of the crossed product $A\rtimes_{\alpha}G$ in terms of the topological data associated to the given action $\alpha:G\to \operatorname{Aut} A$and the bundle structure of the group $C^*$-algebra $C^*(H)$ of $H$.
Rotation, entropy, and equilibrium states
Oliver
Jenkinson
3713-3739
Abstract: For a dynamical system $(X,T)$ and function $f:X\to\mathbb{R} ^d$ we consider the corresponding generalised rotation set. This is the convex subset of $\mathbb{R} ^d$ consisting of all integrals of $f$ with respect to $T$-invariant probability measures. We study the entropy $H(\varrho)$of rotation vectors $\varrho$, and relate this to the directional entropy $\mathcal{H}(\varrho)$ of Geller & Misiurewicz. For $(X,T)$ a mixing subshift of finite type, and $f$ of summable variation, we prove that if the rotation set is strictly convex then the functions $\mathcal{H}$ and $H$ are in fact one and the same. For those rotation sets which are not strictly convex we prove that $\mathcal{H}(\varrho)$ and $H(\varrho)$can differ only at non-exposed boundary points $\varrho$.
Canonical symbolic dynamics for one-dimensional generalized solenoids
Inhyeop
Yi
3741-3767
Abstract: We define canonical subshift of finite type covers for Williams' one-dimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.
Berezin transform on real bounded symmetric domains
Genkai
Zhang
3769-3787
Abstract: Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$is then unitarily equivalent to the restriction to $G$of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.
Crystal bases for $U_{q}(\Gamma(\sigma_{1},\sigma_{2},\sigma_{3}))$
Yi Ming
Zou
3789-3802
Abstract: We construct crystal bases for certain infinite dimensional representations of the $q$-deformation of the Lie superalgebra $\Gamma (\sigma _{1},\sigma _{2}, \sigma _{3})$.
The Serre spectral sequence of a multiplicative fibration
Yves
Félix;
Stephen
Halperin;
Jean-Claude
Thomas
3803-3831
Abstract: In a fibration $\Omega F \overset{\Omega j}{\rightarrow} \Omega X \overset{\Omega \pi}{\rightarrow}\Omega B$ we show that finiteness conditions on $F$ force the homology Serre spectral sequence with $\mathbb{F} _p$-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, $H_*(\Omega X)$ is ``almost'' isomorphic to $H_*(\Omega B)\otimes H_*(\Omega F)$. One consequence is the conclusion that $X$ is elliptic if and only if $B$ and $F$ are.
Correction to ``Relative Completions of Linear Groups over ${\mathbb Z}[t]$ and ${\mathbb Z}[t,t^{-1}]$''
Kevin
P.
Knudson
3833-3834