Determination of all imaginary cyclic quartic fields with class number $2$
Kenneth
Hardy;
Richard H.
Hudson;
David
Richman;
Kenneth S.
Williams
1-55
Abstract: It is proved that there are exactly $8$ imaginary cyclic quartic fields with class number $2$.
Continuous cohomology and real homotopy type
Edgar H.
Brown;
Robert H.
Szczarba
57-106
Abstract: Various aspects of homotopy theory in the category of simplicial spaces are developed. Topics covered include continuous cohomology, continuous de Rham cohomology, the Kan extension condition, the homotopy relation, fibrations, the Serre spectral sequence, real homotopy type and its relation to graded commutative differential algebras over the reals.
Realizing rotation vectors for torus homeomorphisms
John
Franks
107-115
Abstract: We consider the rotation set $\rho (F)$ for a lift $F$ of a homeomorphism $f:{T^2} \to {T^2}$, which is homotopic to the identity. Our main result is that if a vector $v$ lies in the interior of $\rho (F)$ and has both coordinates rational, then there is a periodic point $x \in {T^2}$ with the property that $\displaystyle \frac{{{F^q}({x_0}) - {x_0}}}{q} = v$ where ${x_0} \in {R^2}$ is any lift of $x$ and $q$ is the least period of $x$.
Lie sphere transformations and the focal sets of certain taut immersions
Steven G.
Buyske
117-133
Abstract: We study the images of certain taut or Dupin hypersurfaces, including their focal sets, under Lie sphere transformations (generalizations of conformal transformations of euclidean or spherical space). After the introduction, the method of studying hypersurfaces as Lie sphere objects is developed. In two recent papers, Cecil and Chern use submanifolds of the space of lines on the Lie quadric. Here we use submanifolds of the Lie quadric itself instead. The third section extends the concepts of tightness and tautness to semi-euclidean space. The final section shows that if a hypersurface is the Lie sphere image of certain standard constructions (tubes, cylinders, and rotations) over a taut immersion, the resulting family of curvature spheres is taut in the Lie quadric. The sheet of the focal set will be tight in euclidean space if it is compact. In particular, if a hypersurface in euclidean space is the Lie sphere image of an isoparametric hypersurface each compact sheet of the focal set will be tight.
Peripherally specified homomorphs of knot groups
Dennis
Johnson;
Charles
Livingston
135-146
Abstract: Let $G$ be a group and let $\mu$ and $\lambda$ be elements of $G$. Necessary and sufficient conditions are presented for the solution of the following problem: Is there a knot $K$ in ${S^3}$ and a representation $\rho :{\pi _1}({S^3} - K) \to G$ such that $\rho (m) = \mu $ and $\rho (l) = \lambda$, where $m$ and $l$ are the meridian and longitude of $K$?
Spectral decompositions of one-parameter groups of isometries on Hardy spaces
Dimitri
Karayannakis
147-166
Abstract: Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on Hardy spaces of the disk ${\mathbf{D}}$ and the torus ${{\mathbf{T}}^2}$ are examined; a concrete description of the (pointwise) action of these decompositions is presented, mainly in the parabolic case, leading to a complete description of the action of the partial sum-operators of M. Riesz when carried from ${L^p}({\mathbf{R}})$ to ${H^p}({\mathbf{D}})$, $ 1 < p \leq 2$. The (pointwise) action of the spectral decompositions of these isometric groups on $ {H^p}({{\mathbf{T}}^2})$, $1 < p < \infty$ is also examined and concrete descriptions are derived, mainly in the parabolic case.
Differential-difference operators associated to reflection groups
Charles F.
Dunkl
167-183
Abstract: There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in ${{\mathbf{R}}^n}$. A commutative set of differential-difference operators, each homogeneous of degree $ -1$, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of $ {{\mathbf{R}}^2}$ and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.
Torsion points on abelian \'etale coverings of ${\bf P}\sp 1-\{0,1,\infty\}$
Robert F.
Coleman
185-208
Abstract: Let $X \to {{\mathbf{P}}^1}$ be an Abelian covering of degree $m$ over $ {\mathbf{Q}}({\mu _m})$ unbranched outside 0, $1$ and $\infty$. If the genus of $X$ is greater than $1$ embed $X$ in its Jacobian $J$ in such a way that one of the points above 0, $ 1$ or $\infty$ is mapped to the origin. We study the set of torsion points of $J$ which lie on $X$. In particular, we prove that this set is defined over an extension of $ {\mathbf{Q}}$ unramified outside $6m$. We also obtain information about the orders of these torsion points.
A converse to the mean value property on homogeneous trees
Massimo A.
Picardello;
Wolfgang
Woess
209-225
Abstract: The homogeneous tree ${\mathbf{T}}$ of degree $q + 1\quad (q \geq 2)$ may be considered as a discrete analogue of the open unit disc ${\mathbf{D}}$. On $ {\mathbf{D}}$, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on ${\mathbf{D}}$ having the MVP with respect to a ball with specified radius at each point of ${\mathbf{D}}$ are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on ${\mathbf{T}}$ with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of ${\mathbf{T}}$) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.
Proper holomorphic mappings from the two-ball to the three-ball
J. A.
Cima;
T. J.
Suffridge
227-239
Abstract: We prove that a proper mapping of the two ball in ${\mathbf{C}^{n}}$ into the three ball, which is $ {C^2}$ on the closed two ball is equivalent to one of four normalized polynomial mappings. This improves the known result of Faran. The proof is basic using Taylor expansions.
Torsion points of generic formal groups
Michael
Rosen;
Karl
Zimmermann
241-253
Abstract: Let $F$ be a generic formal group of height $h$ defined over $A = {{\mathbf{Z}}_p}[[{t_1}, \ldots ,{t_{h - 1}}]]$. Let $K$ be the quotient field of $A$. We show the natural map ${\rho _n}:{\text{Gal}}(K(\operatorname{ker} [{p^n}])/K) \to G{L_h}({\mathbf{Z}}/{p^n}{\mathbf{Z}})$ isomorphisms for all $n \ge 1$ provided $p \ne 2$.
Topological entropy of homoclinic closures
Leonardo
Mendoza
255-266
Abstract: In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in $n$-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen's one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.
Excessive measures and the existence of right semigroups and processes
J.
Steffens
267-290
Abstract: Given a resolvent $({U_\alpha })$ on a Lusin space $(E,\mathfrak{E})$, the paper gives necessary and sufficient conditions in terms of the excessive measures that ensure the existence of a right process, resp. a right continuous semigroup, on $(E,\mathfrak{E})$ with resolvent $({U_\alpha })$. Furthermore, a notion of nonbranch points with respect to $({U_\alpha })$ is introduced--also in terms of the excessive measures--and various characterizations are given. They show, in particular, the equivalence of this definition with those introduced and discussed by Engelbert and Wittmann.
Spreading of singularities at the boundary in semilinear hyperbolic mixed problems. II. Crossing and self-spreading
Mark
Williams
291-321
Abstract: The creation of anomalous singularities in solutions to nonlinear hyperbolic equations due to crossing or self-spreading in free space is by now rather well understood. In this paper we study how anomalous singularities are produced in mixed problems for semilinear wave equations $\square u = f(u)$ on the half-space ${\mathbf{R}}_ + ^{n + 1},u \in H_{{\operatorname{loc}}}^s,s > (n + 1)/2$, due to crossing and self-spreading at boundary points. Several phenomena appear in the problems considered here which distinguish spreading at the boundary from spreading in free space: (1) Anomalous singularities of strength $\sim 2s - n/2$ can arise when incoming singularity-bearing rays cross or self-spread at a point on the boundary. A consequence of this, announced in [14], is that the analogue of Beals's $3s$ theorem fails for reflection in second-order mixed problems. Although ${H^r}$ regularity for $r < \sim 3s -n$ propagates along null bicharacteristics in free space, for $r < \sim 2s -n/2$ it does not in general reflect. (2) For nonlinear wave equations in free space, anomalous singular support is never produced by the interaction of fewer than three bicharacteristics, unless self-spreading occurs. However, anomalous singularities can arise when a pair of rays cross at a boundary point. (3) Suppose $ \square u = {u^2}$ and $u \in {C^\infty }$ on the boundary. For certain choices of initial data, anomalous singularities of strength $ \sim 2s - n/2$ arise at the boundary from three sources: interactions of incoming rays with incoming rays, incoming rays with reflected rays, and reflected rays with reflected rays. Singularities produced by the incoming-reflected interactions differ in sign from and are strictly weaker than the other two types, so some cancellations occur. As the incoming rays approach being gliding rays, the difference in strength decreases and hence the cancellations become increasingly significant.
Primary cycles on the circle
Louis
Block;
Ethan M.
Coven;
Leo
Jonker;
Michał
Misiurewicz
323-335
Abstract: We consider cycles, i.e., periodic orbits, of continuous degree one maps of the circle. One cycle forces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second. We characterize primary cycles, i.e., those which force no other cycle of the same period.
Conjugating homeomorphisms to uniform homeomorphisms
Katsuro
Sakai;
Raymond Y.
Wong
337-356
Abstract: Let $H(X)$ denote the group of homeomorphisms of a metric space $X$ onto itself. We say that $h \in H(X)$ is conjugate to $g \in H(X)$ if $ {g = fhf^{-1}}$ for some $f \in H(X)$. In this paper, we study the questions: When is $h \in H(X)$ conjugate to $g \in H(X)$ which is a uniform homeomorphism or can be extended to a homeomorphism $\tilde{g}$ on the metric completion of $ X$ Typically for a complete metric space $X$, we prove that $ h \in H(X)$ is conjugate to a uniform homeomorphism if $H$ is uniformly approximated by uniform homeomorphisms. In case $ X = \mathbf{R}$, we obtain a stronger result showing that every homeomorphism on $\mathbf{R}$ is, in fact, conjugate to a smooth Lipschitz homeomorphis. For a noncomplete metric space $ X$, we provide answers to the existence of $\tilde{g}$ under several different settings. Our results are concerned mainly with infinite-dimensional manifolds.
Translation semigroups and their linearizations on spaces of integrable functions
Annette
Grabosch
357-390
Abstract: Of concern is the unbounded operator $D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}$ which is considered on the Banach space $E$ of Bochner integrable functions on an interval with values in a Banach space $F$. Under the assumption that $\Phi$ is a Lipschitz continuous operator from $E$ to $F$, it is shown that $ {A_{\Phi}}$ generates a strongly continuous translation semigroup ${({T_\Phi }(t))_{t \geq 0}}$. For linear operators $\Phi$ several properties such as essential-compactness, positivity, and irreducibility of the semigroup $ {({T_\Phi }(t))_{t \geq 0}}$ depending on the operator $\Phi$ are studied. It is shown that if $F$ is a Banach lattice with order continuous norm, then ${({T_\Phi }(t))_{t \geq 0}}$ is the modulus semigroup of $ {({T_\Phi }(t))_{t \geq 0}}$. Finally spectral properties of ${A_\Phi}$ are studied and the spectral bound $s({A_\Phi })$ is determined. This leads to a result on the global asymptotic behavior in the case where $\Phi$ is linear and to a local stability result in the case where $\Phi$ is Fréchet differentiable.
Variations of Hodge structure, Legendre submanifolds, and accessibility
James A.
Carlson;
Domingo
Toledo
391-411
Abstract: Variations of Hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain. This distribution has dimension ${h^{2,0}}{h^{1,1}}$ and is nonintegrable for ${h^{2,0}} > 1$. In this case it is known that the dimension of an integral manifold does not exceed $ \frac{1} {2}{h^{2,0}}{h^{1,1}}$. Here we give a new proof, based on an analogy between Griffiths' horizontal differential system of algebraic geometry and the contact system of classical mechanics. We show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic.
Specializations of finitely generated subgroups of abelian varieties
D. W.
Masser
413-424
Abstract: Given a generic Mordell-Weil group over a function field, we can specialize it down to a number field. It has been known for some time that the resulting homomorphism of groups is injective "infinitely often". We prove that this is in fact true "almost always", in a sense that is quantitatively nearly best possible.
Correction to: ``Harmonically immersed surfaces in ${\bf R}\sp n$'' [Trans. Amer. Math. Soc. {\bf 307} (1988), no. 1, 363--372; MR0936822 (89g:53006)]
Gary R.
Jensen;
Marco
Rigoli
425-428