Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts
Peter
Walters
1-31
Abstract: Let $S:X \to X,T:Y \to Y$ be continuous maps of compact metrizable spaces, and let $\pi :X \to Y$ be a continuous surjection with $ \pi \circ S = T \circ \pi$. We investigate the notion of relative pressure, which was introduced by Ledrappier and Walters, and study some maximal relative pressure functions that tie in with relative equilibrium states. These ideas are connected with the notion of compensation function, first considered by Boyle and Tuncel, and we show that a compensation function always exists when $S$ and $T$ are subshifts. A function $F \in C(X)$ is a compensation function if $P(S,F + \phi \circ \pi) = P(T,\phi)\forall \phi \in C(Y)$. When $S$ and $T$ are topologically mixing subshifts of finite type, we relate compensation functions to lifting $ T$-invariant measures to $ S$-invariant measures, obtaining some results of Boyle and Tuncel. We use compensation functions to describe different types of quotient maps $\pi$. An example is given where no compensation function exists.
Isometries for the Legendre-Fenchel transform
Hédy
Attouch;
Roger J.-B.
Wets
33-60
Abstract: It is shown that on the space of lower semicontinuous convex functions defined on ${R^n}$, the conjugation map--the Legendre-Fenchel transform--is an isometry with respect to some metrics consistent with the epi-topology. We also obtain isometries for the infinite dimensional case (Hilbert space and reflexive Banach space), but this time they correspond to topologies finer than the Moscoepi-topology.
Inequalities for the ergodic maximal function and convergence of the averages in weighted $L\sp p$-spaces
F. J.
Martín-Reyes
61-82
Abstract: This paper is concerned with the characterization of those positive functions $w$ such that Hopf's averages associated to an invertible measure preserving transformation $ T$ and a positive function $ g$ converge almost everywhere for every $ f \in {L^p}(w\,d\mu)$. We also study mean convergence when $g$ satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions $(u,w)$ such that the ergodic maximal operator associated to $T$ and $g$ is of weak or strong type with respect to the measures $w\,d\mu$ and $u\,d\mu$.
Desingularizations of plane vector fields
F.
Cano
83-93
Abstract: The singularities of a plane vector field can be reduced under quadratic blowing ups. We describe a control method for the singularities of the vector field which works for ground fields of any characteristic and which has no essential obstruction for generalizing to higher dimensional cases.
The spectrum $(P\wedge{\rm BP}\langle 2\rangle)\sb {-\infty}$
Donald M.
Davis;
David C.
Johnson;
John
Klippenstein;
Mark
Mahowald;
Steven
Wegmann
95-110
Abstract: The spectrum ${(P \wedge {\text{BP}}\langle {\text{2}}\rangle)_{ - \infty }}$ is defined to be the homotopy inverse limit of spectra $ {P_{ - k}} \wedge {\text{BP}}\langle {\text{2}}\rangle $, where ${P_{ - k}}$ is closely related to stunted real projective spaces, and ${\text{BP}}\langle {\text{2}}\rangle$ is formed from the Brown-Peterson spectrum. It is proved that this spectrum is equivalent to the infinite product of odd suspensions of the $2$-adic completion of the spectrum of connective $ K$-theory. An odd-primary analogue is also proved.
Factorial property of a ring of automorphic forms
Shigeaki
Tsuyumine
111-123
Abstract: A ring of automorphic forms is shown to be factorial under some conditions on the domain and on the Picard group. As an application, we show that any divisor on the moduli space $ {\mathfrak{M}_g}$ of curves of genus $g \geqslant 3$ is defined by a single element, and that the Satake compactification of $ {\mathfrak{M}_g}$ is written as a projective spectrum of a factorial graded ring. We find a single element which defines the closure of $ {\mathfrak{M}\prime_4}$ in $ {\mathfrak{M}_4}$ where $ {\mathfrak{M}\prime_4}$ is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in ${\mathbb{P}^3}$.
Insufficiency of Torres' conditions for two-component classical links
M. L.
Platt
125-136
Abstract: Torres has given necessary conditions for a polynomial to be the Alexander polynomial of a two component link. For certain links, additional conditions are necessary. Hillman gave one example for linking number $6$. Here we give examples for all other linking numbers except $0, \pm 1$, and $\pm 2$.
Real hypersurfaces and complex submanifolds in complex projective space
Makoto
Kimura
137-149
Abstract: Let $M$ be a real hypersurface in ${P^n}({\mathbf{C}})$ be the complex structure and $ \xi$ denote a unit normal vector field on $M$. We show that $M$ is (an open subset of) a homogeneous hypersurface if and only if $M$ has constant principal curvatures and $ J\xi$ is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, ${P^m}({\mathbf{C}})$ (totally geodesic), $ {Q^n},{P^1}({\mathbf{C}}) \times {P^n}({\mathbf{C}}),SU(5)/S(U(2) \times U(3))$ and $ SO(10)/U(5)$ are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.
Spanier-Whitehead duality in \'etale homotopy
Roy
Joshua
151-166
Abstract: We construct a $ (\bmod{\text{-}}l)$ Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a $ (\bmod {\text{-}}l)$ Spanier-Whitehead dual.
Paths and cycles in tournaments
Andrew
Thomason
167-180
Abstract: Sufficient conditions are given for the existence of an oriented path with given end vertices in a tournament. As a consequence a conjecture of Rosenfeld is established. This states that if $n$ is large enough, then every non-strongly oriented cycle of order $n$ is contained in every tournament of order $ n$.
Equivariant bundles and cohomology
A.
Kozlowski
181-190
Abstract: Let $G$ be a topological group, $ A$ an abelian topological group on which $G$ acts continuously and $X$ a $G$-space. We define "equivariant cohomology groups" of $X$ with coefficients in $A$, $ H_G^i(X;A)$, for $ i \geq 0$ which generalize Graeme Segal's continuous cohomology of the topological group $G$ with coefficients in $A$. In particular we have $H_G^1(X;A) \simeq$ equivalence classes of principal $(G,A)$-bundles over $X$. We show that when $G$ is a compact Lie group and $A$ an abelian Lie group we have for $i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)$ where $\tau A$ is the sheaf of germs of sections of the bundle $(X \times EG \times A)/G \to (X \times EG)/G$. For $ i = 1$ and the trivial action of $G$ on $A$ this is a theorem of Lashof, May and Segal.
A parametrix for step-two hypoelliptic diffusion equations
Thomas
Taylor
191-215
Abstract: In this paper I construct a parametrix for the hypoelliptic diffusion equations $ (\partial /\partial t - L)u = 0$, where $L = \sum\nolimits_{a = 1}^n {g_a^2}$ and where the $ {g_a}$ are vector fields which satisfy the property that they, together with all of the commutators $ [{g_{a,}}{g_b}]$ for $a < b$, are at each point linearly independent and span the tangent space.
Automorphic images of commutative subspace lattices
K. J.
Harrison;
W. E.
Longstaff
217-228
Abstract: Let $C(H)$ denote the lattice of all (closed) subspaces of a complex, separable Hilbert space $ H$. Let $({\text{AC)}}$ be the following condition that a subspace lattice $\mathcal{F} \subseteq C(H)$ may or may not satisfy: (AC) \begin{displaymath}\begin{array}{*{20}{c}} {\mathcal{F} = \phi (\mathcal{L})\;{\... ...;{\text{lattice}}\;\mathcal{L} \subseteq C(H).} \end{array} \end{displaymath} Then $ \mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{A} \subseteq \mathcal{B}$ for some Boolean algebra subspace lattice $\mathcal{B} \subseteq C(H)$ with the property that, for every $K,L \in \mathcal{B}$, the vector sum $ K + L$ is closed. If $\mathcal{F}$ is finite, then $\mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{F}$ is distributive and $ K + L$ is closed for every $ K,L \in \mathcal{F}$. In finite dimensions $ \mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{F}$ is distributive. Every $\mathcal{F}$ satisfying $({\text{AC}})$ is reflexive. For such $\mathcal{F}$, given vectors $x,y \in H$, the solvability of the equation $Tx = y$ for $T \in \operatorname{Alg}\,\mathcal{F}$ is investigated.
``Almost'' implies ``near''
Robert M.
Anderson
229-237
Abstract: We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.
On excursions of reflecting Brownian motion
Pei
Hsu
239-264
Abstract: We discuss the properties of excursions of reflecting Brownian motion on a bounded smooth domain in ${R^d}$ and give a procedure for constructing the process from the excursions and the boundary process. Our method is computational and can be applied to general diffusion processes with reflecting type boundary conditions on compact manifolds.
On integers free of large prime factors
Adolf
Hildebrand;
Gérald
Tenenbaum
265-290
Abstract: The number $\Psi (x,y)$ of integers $\leq x$ and free of prime factors $> y$ has been given satisfactory estimates in the regions $y \leq {(\log x)^{3/4 - \varepsilon }}$ and $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates $\Psi (x,y)$ uniformly for $x \geq y \geq 2$ within a factor $1 + O((\log y)/(\log x) + (\log y)/y)$. As an application, we derive a simple formula for $\Psi (cx,y)/\Psi (x,y)$, where $1 \leq c \leq y$. We also prove a short interval estimate for $ \Psi (x,y)$.
On a conormal module of smooth set theoretic complete intersections
M.
Boratyński
291-300
Abstract: We prove that $V \subset {\mathbf{A}}_k^n$ ($ V$-smooth) is a set-theoretic complete intersection (stci) if and only if $ V$ imbedded as a zero section of its normal bundle is a stci, we give a characterization of smooth codimension $2$ stci of index $\leq 4$ in terms of their conormal modules.
Chebyshev rank in $L\sb 1$-approximation
András
Kroó
301-313
Abstract: Let ${C_\omega }(K)$ denote the space of continuous functions endowed with the norm ${\smallint _K}\omega \left\vert f \right\vert = {\left\Vert f \right\Vert _\omega },\omega > 0$. In this paper we characterize the subspaces $ {U_n} \subset {C_\omega }(K)$ having Chebyshev rank at most $k\;(0 \leq k \leq n - 1)$ with respect to all bounded positive weights $\omega$. Various applications of main results are also presented.
Blow up near higher modes of nonlinear wave equations
Natalia
Sternberg
315-325
Abstract: This paper is concerned with the instability properties of higher modes of the nonlinear wave equation ${u_{tt}} - \Delta u - f(u) = 0$ defined on a smoothly bounded domain with Dirichlet boundary conditions. It is shown that they are unstable in the sense that in any neighborhood of a higher mode there exists a solution of the given equation which blows up in finite time.
Orbits of the pseudocircle
Judy
Kennedy;
James T.
Rogers
327-340
Abstract: The following theorem is proved. Theorem. The pseudocircle has uncountably many orbits under the action of its homeomorphism group. Each orbit is the union of uncountably many composants.
Inequalities for some maximal functions. II
M.
Cowling;
G.
Mauceri
341-365
Abstract: Let $S$ be a smooth compact hypersurface in ${{\mathbf{R}}^n}$, and let $\mu$ be a measure on $S$, absolutely continuous with respect to surface measure. For $t$ in ${{\mathbf{R}}^ + },{\mu _t}$ denotes the dilate of $ \mu$ by $t$, normalised to have the same total variation as $\mu$: for $f$ in $ \mathcal{S}({{\mathbf{R}}^n}),{\mu ^\char93 }f$ denotes the maximal function $ {\sup _{t > 0}}\vert{\mu _t}\ast f\vert$. We seek conditions on $\mu$ which guarantee that the a priori estimate $\displaystyle \left\Vert \mu^\char93 f\right\Vert _p \leq C\left\Vert f \right\Vert _p, \quad f \in S(\mathbf{R}^n),$ holds; this estimate entails that the sublinear operator ${\mu ^\char93 }$ extends to a bounded operator on the Lebesgue space $ {L^p}({{\mathbf{R}}^n})$. Our methods generalise E. M. Stein's treatment of the "spherical maximal function" [5]: a study of "Riesz operators", $g$-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of $\mu$. These depend on the geometry of $ S$ and the relation between $\mu$ and surface measure on $S$. In particular, we find that there are natural geometric maximal operators limited on ${L^p}({{\mathbf{R}}^n})$ if and only if $p \in (q,\infty ];q$ is some number in $(1,\infty)$, and may be greater than $2$. This answers a question of S. Wainger posed by Stein [6]>.
A critical set with nonnull image has large Hausdorff dimension
Alec
Norton
367-376
Abstract: The question of how complicated a critical set must be to have a nonnull image is answered by relating its Hausdorff dimension to the (Hölder) differentiability of the map. This leads to a new extension of the Morse-Sard Theorem. The main tool is an extended version of Morse's Lemma.
On starshaped rearrangement and applications
Bernhard
Kawohl
377-386
Abstract: A radial symmetrization technique is investigated and new properties are proven. The method transforms functions $ u$ into new functions $ {u^\ast}$ with starshaped level sets and is therefore called starshaped rearrangement. This rearrangement is in general not equimeasurable, a circumstance with some surprising consequences. The method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems. In particular, the Lipschitz continuity of free boundaries can now be easily obtained in a new fashion.
Harmonic analysis on Grassmannian bundles
Robert S.
Strichartz
387-409
Abstract: The harmonic analysis of the Grassmannian bundle of $k$-dimensional affine subspaces of ${{\mathbf{R}}^n}$, as a homogeneous space of the Euclidean motion group, is given explicitly. This is used to obtain the diagonalization of various generalizations of the Radon transform between such bundles. In abstract form, the same technique gives the Plancherel formula for any unitary representation of a semidirect product $G \times V$ ($V$ a normal abelian subgroup) induced from an irreducible unitary representation of a subgroup of the form $H \times W$.
The Fefferman metric and pseudo-Hermitian invariants
John M.
Lee
411-429
Abstract: C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex $n$-space carries a natural conformal Lorentz metric on a circle bundle over the manifold. This paper presents two intrinsic constructions of the metric, valid on an abstract $ {\text{CR}}$ manifold. One is in terms of tautologous differential forms on a natural circle bundle; the other is in terms of Webster's pseudohermitian invariants. These results are applied to compute the connection and curvature forms of the Fefferman metric explicitly.