Model theory of difference fields
Zoé
Chatzidakis;
Ehud
Hrushovski
2997-3071
Abstract: A difference field is a field with a distinguished automorphism $\sigma$. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is $0$.
Forcing minimal extensions of Boolean algebras
Piotr
Koszmider
3073-3117
Abstract: We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk's method more flexible, obtaining, for every cardinal $\lambda$ of uncountable cofinality, a consistent example of a Boolean algebra $A_{\lambda }$ whose every infinite homomorphic image is of cardinality $\lambda$ and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight $\lambda$). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra $A$ can be strictly less than the minimal size of a homomorphic image of $A$, answering a question of J. D. Monk. 2) We prove that for every cardinal of uncountable cofinality it is consistent that $2^{\omega }=\lambda$ and both $A_{\lambda }$ and $A_{\omega _{1}}$ exist. 3) By combining these algebras we obtain many examples that answer questions of J.D. Monk. 4) We prove the consistency of MA + $\neg$CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel$'$skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character. 5) We prove that for any cardinal $\lambda$ of uncountable cofinality it is consistent that there is a countably tight Boolean algebra $A$ with a distinguished ultrafilter $\infty$ such that for every $a\not \ni \infty$ the algebra $A|a$ is countable and $\infty$ has hereditary character $\lambda$.
The maximality of the core model
E.
Schimmerling;
J.
R.
Steel
3119-3141
Abstract: Our main results are: 1) every countably certified extender that coheres with the core model $K$ is on the extender sequence of $K$, 2) $K$ computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of $K$, 4) (joint with W. J. Mitchell) $K\|\kappa$ is universal for mice of height $\le\kappa$ whenever $\kappa\geq\aleph _2$, 5) if there is a $\kappa$ such that $\kappa$ is either a singular countably closed cardinal or a weakly compact cardinal, and $\square _\kappa^{<\omega}$ fails, then there are inner models with Woodin cardinals, and 6) an $\omega$-Erdös cardinal suffices to develop the basic theory of $K$.
Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights
Albrecht
Böttcher;
Yuri
I.
Karlovich
3143-3196
Abstract: We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space $H^p(\Gamma,\omega)$ in case $1<p<\infty$, $\Gamma$ is a Carleson Jordan curve and $\omega$ is a Muckenhoupt weight in $A_p(\Gamma)$. Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve $\Gamma$ and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve $\Gamma$ is nice and $\omega$ is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.
Bilipschitz homogeneous Jordan curves
Manouchehr
Ghamsari;
David
A.
Herron
3197-3216
Abstract: We characterize bilipschitz homogeneous Jordan curves by utilizing quasihomogeneous parameterizations. We verify that rectifiable bilipschitz homogeneous Jordan curves satisfy a chordarc condition. We exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower Hausdorff density zero. We examine homeomorphisms between Jordan curves.
A theorem on zeta functions associated with polynomials
Minking
Eie;
Kwang-Wu
Chen
3217-3228
Abstract: Let $\beta =(\beta _{1},\ldots ,\beta _{r})$ be an $r$-tuple of non-negative integers and $P_{j}(X)$ $(j=1,2,\ldots ,n)$ be polynomials in ${\mathbb{R}}[X_{1},\ldots ,X_{r}]$ such that $P_{j}(n)>0$ for all $n\in {\mathbb{N}}^{r}$ and the series \begin{equation*}\sum _{n\in {\mathbb{N}}^{r}} P_{j}(n)^{-s}\end{equation*} is absolutely convergent for Re $s>\sigma _{j}>0$. We consider the zeta functions \begin{equation*}Z(P_{j},\beta ,s)=\sum _{n\in{\mathbb{N}}^{r}}n^{\beta} P_{j}(n)^{-s},\quad \text{Re} s>|\beta |+\sigma _{j}, \quad 1\leq j\leq n.\end{equation*} All these zeta functions $Z(\prod ^{n}_{j=1} P_{j},\beta ,s)$ and $Z(P_{j},\beta ,s)\quad (j=1,2,\ldots ,n)$ are analytic functions of $s$ when Re$\, s$ is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that \begin{equation*}Z(\prod _{j=1}^{n} P_{j},\beta ,0)=\frac{1}{n} \sum _{j=1}^{n} Z(P_{j},\beta ,0).\end{equation*} As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.
Deformations of dihedral 2-group extensions of fields
Elena
V.
Black
3229-3241
Abstract: Given a $G$-Galois extension of number fields $L/K$ we ask whether it is a specialization of a regular $G$-Galois cover of $\mathbb{P}^{1}_{K}$. This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral $2$-groups under certain assumptions on the base field $k$. We also show that dihedral groups of order $8$ and $16$ have generic extensions over any base field $k$ with characteristic different from $2$.
Knot invariants from symbolic dynamical systems
Daniel
S.
Silver;
Susan
G.
Williams
3243-3265
Abstract: If $G$ is the group of an oriented knot $k$, then the set $\operatorname{Hom} (K, \Sigma )$ of representations of the commutator subgroup $K = [G,G]$ into any finite group $\Sigma$ has the structure of a shift of finite type $\Phi _{\Sigma }$, a special type of dynamical system completely described by a finite directed graph. Invariants of $\Phi _{\Sigma }$, such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When $\Sigma$ is abelian, $\Phi _{\Sigma }$ gives information about the infinite cyclic cover and the various branched cyclic covers of $k$. Similar techniques are applied to oriented links.
Hardy spaces and a Walsh model for bilinear cone operators
John
E.
Gilbert;
Andrea
R.
Nahmod
3267-3300
Abstract: The study of bilinear operators associated to a class of non-smooth symbols can be reduced to ther study of certain special bilinear cone operators to which a time frequency analysis using smooth wave-packets is performed. In this paper we prove that when smooth wave-packets are replaced by Walsh wave-packets the corresponding discrete Walsh model for the cone operators is not only $L^{p}$-bounded, as Thiele has shown in his thesis for the Walsh model corresponding to the bilinear Hilbert transform, but actually improves regularity as it maps into a Hardy space. The same result is expected to hold for the special bilinear cone operators.
An estimate for a first-order Riesz operator on the affine group
Peter
Sjögren
3301-3314
Abstract: On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian $\Delta$. For a certain right-invariant vector field $X$, we prove that the first-order Riesz operator $X\Delta^{-1/2}$ is of weak type (1, 1) with respect to the left Haar measure of the group. This operator is therefore also bounded on $L^p, \; 1<p\leq 2$. Locally, the operator is a standard singular integral. The main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation.
Numeration systems and Markov partitions from self similar tilings
Brenda
Praggastis
3315-3349
Abstract: Using self similar tilings we represent the elements of $\mathbb{R}^n$ as digit expansions with digits in $\mathbb{R}^n$ being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.
Invariance principles and Gaussian approximation for strictly stationary processes
Dalibor
Volný
3351-3371
Abstract: We show that in any aperiodic and ergodic dynamical system there exists a square integrable process $(f\circ T^{i})$ the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For $(f\circ T^{i})$ both weak and strong invariance principles hold.
Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures
Naoya
Sumi
3373-3385
Abstract: We consider the $C^{2}$ set of $C^{2}$ diffeomorphisms of the 2-torus $\mathbb{T}^{2}$, provided the conditions that the tangent bundle splits into the directed sum $T\mathbb{T}^{2}=E^{s}\oplus E^{u}$ of $Df$-invariant subbundles $E^{s}$, $E^{u}$ and there is $0<\lambda <1$ such that $\Vert Df|_{E^{s}}\Vert <\lambda$ and $\Vert Df|_{E^{u}}\Vert \ge 1$. Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the $C^{2}$ set has no SBR measures. This is related to a result of Hu-Young.
Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property (T)
Yehuda
Shalom
3387-3412
Abstract: Let $k$ be any locally compact non-discrete field. We show that finite invariant measures for $k$-algebraic actions are obtained only via actions of compact groups. This extends both Borel's density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for $k$-algebraic actions, finitely additive finite invariant measures are obtained only via actions of amenable groups. This gives a new criterion for Zariski density of subgroups and is shown to have representation theoretic applications. The main one is to Kazhdan's property $(T)$ for algebraic groups, which we investigate and strengthen.
On modules of bounded multiplicities for the symplectic algebras
D.
J.
Britten;
F.
W.
Lemire
3413-3431
Abstract: Simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product. Also, it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible.
Gorenstein space with nonzero evaluation map
H.
Gammelin
3433-3440
Abstract: Let $(A,d)$ be a differential graded algebra of finite type, if $H^*(A)$ is a Gorenstein graded algebra, then so is $A$. The purpose of this paper is to prove the converse under some mild hypotheses. We deduce a new characterization of Poincaré duality spaces as well as spaces with a nonzero evaluation map.