Stability of multiple-pulse solutions
Björn
Sandstede
429-472
Abstract: In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of $N$-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the $N$-pulses. As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many $N$-pulses bifurcate for any fixed $N>1$. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and $N-1$ in the right half plane can be prescribed.
Comparison theorems and orbit counting in hyperbolic geometry
Mark
Pollicott;
Richard
Sharp
473-499
Abstract: In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both ``thermodynamic'' ergodic theory and the automaton associated to strongly Markov groups.
Lévy processes in semisimple Lie groups and stability of stochastic flows
Ming
Liao
501-522
Abstract: We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.
Asymptotics for minimal discrete energy on the sphere
A.
B. J.
Kuijlaars;
E.
B.
Saff
523-538
Abstract: We investigate the energy of arrangements of $N$ points on the surface of the unit sphere $S^d$ in $\mathbf{R}^{d+1}$ that interact through a power law potential $V = 1/r^s ,$ where $s > 0$ and $r$ is Euclidean distance. With $\mathcal{E}_d(s,N)$ denoting the minimal energy for such $N$-point arrangements we obtain bounds (valid for all $N$) for $\mathcal{E}_d(s,N)$ in the cases when $0 < s < d$ and $2 \leq d < s$. For $s = d$, we determine the precise asymptotic behavior of $\mathcal{E}_d(d,N)$ as $N \rightarrow \infty$. As a corollary, lower bounds are given for the separation of any pair of points in an $N$-point minimal energy configuration, when $s \geq d \geq 2$. For the unit sphere in $\mathbf{R}^3$ $(d = 2)$, we present two conjectures concerning the asymptotic expansion of $\mathcal{E}_2(s,N)$ that relate to the zeta function $\zeta _L(s)$ for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of $\zeta _L(s)$ when $0 < s < 2$ (the divergent case).
Extremal vectors and invariant subspaces
Shamim
Ansari;
Per
Enflo
539-558
Abstract: For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator $K$ some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with $K$ and that for any normal operator $N$, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with $N$. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if $T$ belongs to a certain class ${\mathcal{C}}$ of operators, then the sequence of such vectors converges in norm, and that if $T$ belongs to a subclass of ${\mathcal{C}}$, then the norm limit is cyclic.
Relativity of the spectrum and discrete groups on hyperbolic spaces
N.
Mandouvalos
559-569
Abstract: We give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group.
Hyperbolic groups and free constructions
O.
Kharlampovich;
A.
Myasnikov
571-613
Abstract: It is proved that the property of a group to be hyperbolic is preserved under HHN-extensions and amalgamated free products provided the associated (amalgamated) subgroups satisfy certain conditions. Some more general results about the preservation of hyperbolicity under graph products are also obtained. Using these results we describe the $\mathbf{Q}$-completion $(\mathbf{Q}$ is the field of rationals) $G^{\mathbf{Q}}$ of a torsion-free hyperbolic group $G$ as a union of an effective chain of hyperbolic subgroups, and solve the conjugacy problem in $G^{\mathbf{Q}}$.
The approximate functional formula for the theta function and Diophantine Gauss sums
E.
A.
Coutsias;
N.
D.
Kazarinoff
615-641
Abstract: We consider the polygonal lines in the complex plane $\Bbb{C}$ whose $N$-th vertex is defined by $S_N = \sum _{n=0}^{N\,'} \exp(i\omega \pi n^2)$ (with $\omega \in \Bbb{R}$), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small $\omega$, to Cornu spirals (C-spirals), we prove the precise renormalization formula \begin{equation}\begin{split} &\left| \sum _{k=0}^{N}\,' \exp(i\omega \pi k^2) -\frac{\exp(sgn(\omega )i\pi /4)}{\sqrt{|\omega |}} \sum _{k=0}^n \,' \exp(-i\frac{\pi}{\omega} k^2)\right| &\qquad\leq C \left|\frac{\omega N - n}{\omega}\right|, \ 0<|\omega | <1, \end{split} \end{equation} where $N=[[n/\omega]]$, the nearest integer to $n/\omega$ and $1<C<3.14$ . This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map \begin{equation}\omega \rightarrow -\frac{1}{\omega} \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation} whose orbits are analyzed by expressing $\omega$ as an even continued fraction.
Derivations, isomorphisms, and second cohomology of generalized Witt algebras
Dragomir
Z. DJ
Okovic;
Kaming
Zhao
643-664
Abstract: Generalized Witt algebras, over a field $F$ of characteristic $0$, were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras $W=W(A,T,\varphi)$ over $F$, where the ingredients are an abelian group $A$, a vector space $T$ over $F$, and a map $\varphi:T\times A\to K$ which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra $W=$$W(A,T,\varphi)$, with the right kernel of $\varphi$ being $0$, are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group $H^2(W,F)$ for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.
On Non-hyperbolic Quasi-convex Spaces
Rafael
Oswaldo
Ruggiero
665-687
Abstract: We show that if the universal covering of a compact Riemannian manifold with no conjugate points is a quasi-convex metric space then the following assertion holds: Either the universal covering of the manifold is a hyperbolic geodesic space or it contains a quasi-isometric immersion of $Z\times Z$.
Comparing Heegaard splittings -the bounded case
Hyam
Rubinstein;
Martin
Scharlemann
689-715
Abstract: In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If $p \leq q$ are the genera of two splittings, then there is a common stabilization of genus $5p + 8q - 9$. Here we show how to obtain similar results even when the 3-manifold has boundary.
Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
Feliks
Przytycki
717-742
Abstract: We prove that for every rational map on the Riemann sphere $f:\overline{\mathbb{C}} \to \overline{\mathbb{C}}$, if for every $f$-critical point $c\in J$ whose forward trajectory does not contain any other critical point, the growth of $|(f^{n})'(f(c))|$ is at least of order $\exp Q \sqrt n$ for an appropriate constant $Q$ as $n\to \infty$, then $\operatorname{HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname{HD} (J)$. Here $\operatorname{HD}_{\operatorname {ess}} (J)$ is the so-called essential, dynamical or hyperbolic dimension, $\operatorname{HD} (J)$ is Hausdorff dimension of $J$ and $\alpha _{0}$ is the minimal exponent for conformal measures on $J$. If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of $J$ also coincides with $\operatorname{HD}(J)$. We prove ergodicity of every $\alpha$-conformal measure on $J$ assuming $f$ has one critical point $c\in J$, no parabolic, and $\sum _{n=0}^{\infty }|(f^{n})'(f(c))|^{-1} <\infty$. Finally for every $\alpha$-conformal measure $\mu$ on $J$ (satisfying an additional assumption), assuming an exponential growth of $|(f^{n})'(f(c))|$, we prove the existence of a probability absolutely continuous with respect to $\mu$, $f$-invariant measure. In the Appendix we prove $\operatorname{HD}_{\operatorname {ess}} (J)=\operatorname {HD} (J)$ also for every non-renormalizable quadratic polynomial $z\mapsto z^{2}+c$ with $c$ not in the main cardioid in the Mandelbrot set.
$L^p$ and operator norm estimates for the complex time heat operator on homogeneous trees
Alberto
G.
Setti
743-768
Abstract: Let $\mathfrak{X}$ be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator ${\mathcal{H}}_{\zeta }$ induced by the natural Laplace operator on $\mathfrak{X}$. We prove comparable upper and lower bounds for the $L^{p}$ norms of its convolution kernel $h_{\zeta }$ and derive precise estimates for the $L^{p}\text{--}L^{r}$ operator norms of ${\mathcal{H}}_{\zeta }$ for $\zeta$ belonging to the half plane $\text{Re}\,\zeta \geq 0.$ In particular, when $\zeta$ is purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on $\mathfrak{X}$.
Contiguous relations, continued fractions and orthogonality
Dharma
P.
Gupta;
David
R.
Masson
769-808
Abstract: We examine a special linear combination of balanced very-well-poised ${_{10} \phi _{9}}$ basic hypergeometric series that is known to satisfy a transformation. We call this $\Phi$ and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for $\Phi$ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's $q$-analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new $q$-series identities are also obtained. One is an important three-term transformation for $\Phi$'s which generalizes all the known two- and three-term ${_{8} \phi _{7}}$ transformations. Others are new and unexpected quadratic identities for these very-well-poised ${_{8} \phi _{7}}$'s.
Recognizing constant curvature discrete groups in dimension 3
J.
W.
Cannon;
E.
L.
Swenson
809-849
Abstract: We characterize those discrete groups $G$ which can act properly discontinuously, isometrically, and cocompactly on hyperbolic $3$-space ${\mathbb H}^3$ in terms of the combinatorics of the action of $G$ on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the $2$-sphere.
Erratum to ``Orthogonal calculus''
Michael
S.
Weiss
851-855