Tiling the projective foliation space of a punctured surface
Lee
Mosher
1-70
Abstract: There is a natural way to associate, to each ideal triangulation of a punctured surface a cell decomposition of the projective foliation space of the punctured surface.
The automorphism group of a shift of finite type
Mike
Boyle;
Douglas
Lind;
Daniel
Rudolph
71-114
Abstract: Let $({X_T},{\sigma _T})$ be a shift of finite type, and $G = \operatorname{aut} ({\sigma _T})$ denote the group of homeomorphisms of ${X_T}$ commuting with $ {\sigma _T}$. We investigate the algebraic properties of the countable group $ G$ and the dynamics of its action on ${X_T}$ and associated spaces. Using "marker" constructions, we show $G$ contains many groups, such as the free group on two generators. However, $G$ is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on $n$ coordinates leads to a new and nontrivial topological invariant of $ {\sigma _T}$ whose exact value is not known. We prove that, modulo a few points of low period, $G$ acts transitively on the set of points with least ${\sigma _T}$-period $n$. Using $p$-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of $G$ on the dimension group of ${\sigma _T}$ is investigated. We show there are no proper infinite compact $G$-invariant sets. We give a complete characterization of the $G$-orbit closure of a continuous probability measure, and deduce that the only continuous $G$-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.
Infinitesimally rigid polyhedra. II. Modified spherical frameworks
Walter
Whiteley
115-139
Abstract: In the first paper, Alexandrov's Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz's Theorem, and a related coordinatization of the proper realizations of a $3$-connected spherical polyhedron. Sample results show that: (a) a single $k$-gonal block and a $k$-gonal hole yield static rigidity if and only if the block and hole are $k$-connected in a vertex sense; and (b) a $ 4$-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.
On the second fundamental theorem of Nevanlinna
Arturo
Fernández Arias
141-163
Abstract: It is shown that a condition on the size of the exceptional set in the second fundamental theorem of Nevanlinna cannot be improved. The method is based on a construction of Hayman and also makes use of a quantitative version of a result of F. Nevanlinna about the growth of the characteristic function of a meromorophic function omitting a finite number of points
Weyl groups and the regularity properties of certain compact Lie group actions
Eldar
Straume
165-190
Abstract: The geometric weight system of a $G$-manifold $X$ (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given $G$-weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the $ G$-weight pattern must in fact coincide with that of a simple representation. This in turn implies that $X$ is (orthogonally) modeled on the linear $ G$-space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of $G$-manifolds, $G$ a classical group. In this family the weight patterns are of $2$-parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.
Estimates for $(\overline\partial-\mu\partial)\sp {-1}$ and Calder\'on's theorem on the Cauchy integral
Stephen W.
Semmes
191-232
Abstract: One can view the Cauchy integral operator as giving the solution to a certain $\overline \partial$ problem. If one has a quasiconformal mapping on the plane that takes the real line to the curve, then this $\bar \partial$ problem on the curve can be pulled back to a $ \bar \partial - \mu \partial$ problem on the line. In the case of Lipschitz graphs (or chordarc curves) with small constant, we show how a judicial choice of q.c. mapping and suitable estimates for $ \bar \partial - \mu \partial$ gives a new approach to the boundedness of the Cauchy integral. This approach has the advantage that it is better suited to related problems concerning ${H^\infty }$ than the usual singular integral methods. Also, these estimates for the Beltrami equation have application to quasiconformal and conformal mappings, taken up in a companion paper.
Quasiconformal mappings and chord-arc curves
Stephen W.
Semmes
233-263
Abstract: Given a quasiconformal mapping $\rho$ on the plane, what conditions on its dilatation $ \mu$ guarantee that $\rho ({\mathbf{R}})$ is rectifiable and $ \rho {\vert _{\mathbf{R}}}$ is locally absolutely continuous? We show in this paper that if $\mu$ satisfies certain quadratic Carleson measure conditions, with small norm, then $\rho ({\mathbf{R}})$ is a chord-arc curve with small constant, and $\rho (x) = \rho (0) + \int_0^x {{e^{a(t)}}dt}$ for $ x \in {\mathbf{R}}$, with $ a \in \operatorname{BMO}$ having small norm. Conversely, given any such map from ${\mathbf{R}} \to {\mathbf{C}}$, we show that it has an extension to $ {\mathbf{C}}$ with the right kind of dilatation. Similar results hold with ${\mathbf{R}}$ replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.
Elliptic and parabolic BMO and Harnack's inequality
Hugo
Aimar
265-276
Abstract: We give a generalization of the John-Nirenberg lemma which can be applied to prove ${A_2}$ type conditions for small powers of positive solutions of elliptic and parabolic, degenerate and nondegenerate operators.
A Brouwer translation theorem for free homeomorphisms
Edward E.
Slaminka
277-291
Abstract: We prove a generalization of the Brouwer Translation Theorem which applies to a class of homeomorphisms (free homeomorphisms) which admit fixed points, but retain a dynamical property of fixed point free orientation preserving homeomorphsims. That is, if $h:{M^2} \to {M^2}$ is a free homeomorphism where $ {M^2}$ is a surface, then whenever $D$ is a disc and $h(D) \cap D = \emptyset $, we have that $ {h^n}(D) \cap D = \emptyset$ for all $n \ne 0$. Theorem. Let $h$ be a free homeomorphism of $ {S^2}$, the two-sphere, with finite fixed point set $F$. Then each $p \in {S^2} - F$ lies in the image of an embedding ${\phi _p}:({R^2},\,0) \to ({S^2} - F,\,p)$ such that: (i) $h{\phi _p} = {\phi _p}\tau $, where $\tau (z) = z + 1$ is the canonical translation of the plane, and (ii) the image of each vertical line under ${\phi _p}$ is closed in ${S^2} - F$.
There is no exactly $k$-to-$1$ function from any continuum onto $[0,1]$, or any dendrite, with only finitely many discontinuities
Jo W.
Heath
293-305
Abstract: Katsuura and Kellum recently proved [8] that any (exactly) $ k$-to$1$ function from $[0,\,1]$ onto $[0,\,1]$ must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly) $k$-to-$1$ function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.
A truncated Gauss-Kuz\cprime min law
Doug
Hensley
307-327
Abstract: The transformations $ {T_n}$ which map $x \in [0,\,1)$ onto 0 (if $x \leqslant 1/(n + 1)$), and to $\{ 1/x\}$ otherwise, are truncated versions of the continued fraction transformation $T:x \to \{ 1/x\}$ (but $0 \to 0$). An analog to the Gauss-Kuzmin result is obtained for these ${T_n}$, and is used to show that the Lebesgue measure of $T_n^{ - k}\{ 0\} $ approaches $ 1$ exponentially. From this fact is obtained a new proof that the ratios $ \nu /k$, where $ \nu$ denotes any solution of $ {\nu ^2} \equiv - 1\bmod k$, are uniformly distributed $\bmod 1$ in the sense of Weyl.
The first case of Fermat's last theorem is true for all prime exponents up to $714,591,416,091,389$
Andrew
Granville;
Michael B.
Monagan
329-359
Abstract: We show that if the first case of Fermat's Last Theorem is false for prime exponent $p$ then ${p^2}$ divides ${q^p} - q$ for all primes $q \leqslant 8q$. As a corollary we state the theorem of the title.
An approach to homotopy classification of links
J. P.
Levine
361-387
Abstract: A reformulation and refinement of the $ \overline \mu$-invariants of Milnor are used to give a homotopy classification of $ 4$ component links and suggest a possible general homotopy classification. The main idea is to use the (reduced) group of a link and its "geometric" automorphisms to define the precise indeterminacy of these invariants.
The cohomology representation of an action of $C\sb p$ on a surface
Peter
Symonds
389-400
Abstract: When a finite group $ G$ acts on a surface $ S$, then ${H^1}(S;\,{\mathbf{Z}})$ posseses naturally the structure of a $ {\mathbf{Z}}G$-module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.
Testing analyticity on rotation invariant families of curves
Josip
Globevnik
401-410
Abstract: Let $\Gamma \subset C$ be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let $\Omega$ be the annulus obtained by rotating $ \Gamma$ around the origin. We characterize the curves $\Gamma$ with the property that if $f \in C(\Omega )$ is analytic on $s\Gamma$ for every $s$, $\vert s\vert = 1$, then $f$ is analytic in Int $\Omega$.
Fixed points of arc-component-preserving maps
Charles L.
Hagopian
411-420
Abstract: The following classical problem remains unsolved: If $ M$ is a plane continuum that does not separate the plane and $f$ is a map of $M$ into $M$, must $f$ have a fixed point? We prove that the answer is yes if $f$ maps each arc-component of $ M$ into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author's theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.
Fonctions sph\'eriques des espaces sym\'etriques compacts
Jean-Louis
Clerc
421-431
Abstract: An integral formula, similar to Harish-Chandra's formula for spherical functions on a noncompact Riemannian symmetric space $G/K$ is given for the spherical functions of the compact dual $U/K$. As a consequence, an asymptotic expansion, as the parameter tends to infinity, is obtained, by using the (complex) stationary phase method. RÉSUMÉ. On démontre une formule intégrale pour les fonctions sphériques d'un espace symétrique de type compact $U/K$, analogue de la formule d'Harish-Chandra pour le dual non-compact $G/K$. En conséquence on obtient un équivalent asymptotique lorsque le paramètre tend vers l'infini, en utilisant la méthode de la phase stationnaire complexe.