Schiffer variation of complex structure and coordinates for Teichmüller spaces

Schiffer variation of complex structure on a Riemann surfaceX ₀ is achieved by punching out a parametric disc \(\bar D\) fromX ₀ and replacing it by another Jordan domain whose boundary curve is a holomorphic image of \(\partial \bar D\) . This change of structure depends on a complex parameter ε which determines the holomorphic mapping function around \(\partial \bar D\) . It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmüller spaceT(X ₀), (or reduced Teichmüller spaceT ^#(X₀)). For compactX ₀ this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates. Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmüller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X ₀) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.     

Teichmüller spaces for pointed Fuchsian groups

Let T(G) be the Teichmüller space of a Fuchsian group G and T(G) be the pointed Teichmüller space of a corresponding pointed Fuchsian group G.We will discuss the existence of holomorphic sections of the projection from the space M(G) of Beltrami coefficients for G to T(G) and of that from T(G) to T(G) as well.We will also study the biholomorphic isomorphisms between two pointed Teichmüller spaces.     

On angles in Teichmüller spaces

We discuss the existence of the angle between two curves in Teichmüller spaces and show that, in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length while its three angles are equal to any given three possibly different numbers from 0 to $\pi $ . This implies that the sum of three angles of a geodesic triangle may be equal to any given number from 0 to $3\pi $ in an infinite dimensional Teichmüller space.     

Length spectra and the Teichmüller metric for surfaces with boundary

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε ₀-relative \({\epsilon}\)-thick parts”, and whose definition depends on the choice of some positive constants ε ₀ and \({\epsilon}\). Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.     

Random coverings of the circle with i.i.d. centers

Consider the random intervals In(ω):=(ωn-ln/2,ωn+ln/2)(mod 1) with their centers ωn being i.i.d.but not necessary uniformly distributed on the circle T = R /Z and with their lengths decreasing to zero.Using the dimension theory in dynamical systems,we give conditions on which the circle is finitely or infinitely often covered by intervals In(ω)}n≥1.     

On the representation of measurable functions by martingales

Пусть (gW, ?,P) - вероятност ное пространство, ?₁??₂?...?? _n ?...,? _n ?? -последовательност ь σ-алгебр и ?_∞ - порожден ная ими минимальная σ-алгебра. В статье указано необ ходимое и достаточно е условие на последовательность σ-алгебр {? _n }, при выполнении кото рого для любой ?_∞-измер имой функцииf(x) существует ряд \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) центрированных отн осительно {? _n } функций {? _n } _n=1 ^∞ такой, что

1. \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) абсолютно почти вс юду сходится кf(x) на множестве {x: ¦f(x)¦<+∞};
2. \(\mathop \sum \limits_{n = 1}^\infty \varphi _n (x)\) сходится по мере кf(x) на множестве {х: ¦f(х)¦=+∞ }.

Полученные результа ты представляют обоб щения и усиления доказанных ранее теорем Р. Ганди и Г. Ламба о пре дставлении ?_∞-измерим ых функций мартингалам и {? _n ,? _n } (см. [1] и [2]).     

A note on the density of a subset of all integrable holomorphic quadratic differentials

Let T0(Δ) be the subset of the universal Teichm¨uller space, which consists all of the elements with boundary dilatation 1. Let SQ(Δ) be the unit ball of the space Q(Δ) of all integrable holomorphic quadratic differentials on the unit disk Δ and Q0(Δ) be defined as Q0(Δ) = {? ∈ SQ(Δ) : there exists a k ∈(0, 1) such that [kˉ? |?|] ∈ T0(Δ)}. In this paper, we show that Q0(Δ) is dense in SQ(Δ).     

Computing Teichmüller Maps Between Polygons

By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle-preserving manner). However, when this map is extended to the boundary, it need not necessarily map the vertices of P to those of Q. For many applications, it is important to find the “best” vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps. There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a \((1+\varepsilon )\)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps. This paper solves the problem of finding a finite-time procedure for approximating Teichmüller maps in the continuous setting. Our construction is via an iterative procedure that is proven to converge in \(O(\text {poly}(1/\varepsilon ))\) iterations to a map whose dilatation is at most \(\varepsilon \) more than that of the Teichmüller map, for any \(\varepsilon >0\). We reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete (1) compositions and (2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines, we provide an approximation algorithm with an additive error of at most \(\varepsilon \).     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field $$E_\phi:= \{x\in I: \|T_\beta^nx - x\| < \|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\}$$ and $$F_\phi:=\{x\in I: \|T_\beta^nx-x_0\|<\|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\},$$ where x ₀ is any fixed point in I and ${\phi}$ is any positive function defined on ${\mathbb{N}}$ with ${\phi(n)\to\infty}$ as n → ∞. We completely determine the Hausdorff dimensions of these sets: $$\dim_{\rm H} E_{\phi}=\dim_{\rm H}F_\phi=\frac{1}{1+\liminf\limits_{n\to\infty}\frac{\phi (n)}{n}}.$$      

Distance in the curve graph

We estimate the distance in the curve graph of a surface $S$ of finite type up to a fixed multiplicative constant using Teichmüller geodesics.     

Moment multipliers for power series

По определению после довательность {μ _n пр инадлежит классуG _s , если звезда М иттагЛеффлера произвольного степе нного ряда (1) $$\mathop \sum \limits_0^\infty a_n z^n , \mathop {lim sup}\limits_{n \to \infty } \left| {a_n } \right|^{1/n}< \infty $$ , совпадает со звёздам и Миттаг-Леффлера сте пенных рядов $$\mathop \sum \limits_0^\infty \mu _n a_n z^n ,\mathop \sum \limits_0^\infty \mu _n^{ - 1} a_n z^n $$ . В работе установлены следующие утвержден ия Теорема 1.Для произво льной последователь ности ? _n с условиями $$0< \varphi _n< 1,\mathop {lim}\limits_{n \to \infty } \varphi _n = 0,\mathop {lim}\limits_{n \to \infty } \varphi _n^{1/n} = 1$$ существует неубываю щая функция χ(t) такая, ч то моменты \(\mu _n = \int\limits_0^1 {t^n d\chi (t)} \) удовлетворяют условию 0<μ_nn звезда М иттаг-Леффлера любог о ряда (1) совпадает со звездой МиттагЛеффлера степенных рядов . Теорема 2. Для произвол ьной неотрицательно й последовательности {а_n} с условием {a _n } и для любой последов ательности {?_n} для к оторой 0n<1, \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n = 0\) сущест вуютπ={π _n }∈G _s и последовательнос ть {п_i} такие, что a_nμ_n≦1 (n≧n₀), \(a_{n_i } \mu _{\mu _i } \geqq exp( - \varepsilon _{n_i } )\) (i=1, 2, ...) и при эmom звезда Миттаг-Леффлера ряда (1) совпа дает со звездой Миттаг- Леффлера степенных р ядов .     

Estimate of a polynomial in generalized exponential functions in the convex hull of two domains

Пусть $$f_n (z) = \exp \{ \lambda _n z\} [1 + \psi _n (z)], n \geqq 1$$ гдеψ _n (z) — регулярны в н екоторой односвязно й областиS, λ _n — нули целой функц ии экспоненциальног о ростаL(λ) с индикатрис ой ростаh(?), причем $$|L\prime (\lambda _n )| > C(\delta )\exp \{ [h(\varphi _n ) - \varepsilon ]|\lambda _n |\} \varphi _n = \arg \lambda _n , \forall \varepsilon > 0$$ . Предположим, что на лю бом компактеK?S $$|\psi _n (z)|< Aq^{|\lambda |_n } , a< q< 1, n \geqq 1$$ гдеA иq зависит только отK. Обозначим через \(\bar D\) со пряженную диаграмму функцииL(λ), через \(\bar D_\alpha \) — смещение. \(\bar D\) на векторα. Рассмотр им множестваD ₁ иD ₂ так ие, чтоD ₁ иD ₂ и их вьшуклая обо лочкаE принадлежатS. Пусть \(\bar D_{\alpha _1 } \subset D_1 , \bar D_{\alpha _2 } \subset D_2 \) Доказывается, что сущ ествует некоторая об ластьG?E такая, что \(\mathop \cup \limits_{\alpha \in [\alpha _1 ,\alpha _2 ]} \bar D_\alpha \subset G\) и дляz∈G верна оценка $$\sum\limits_{v = 1}^n {|a_v f_v (z)|} \leqq B\max (M_1 ,M_2 ), M_j = \mathop {\max }\limits_{t \in \bar D_j } |\sum\limits_{v = 1}^n {a_v f_v (t)} |$$ , где константаB не зав исит от {a _v }.     

Weak and strong convergence to common fixed points of non-self nonexpansive mappings

SupposeK is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach spaceE withP as a nonexpansive retraction. LetT ₁,T ₂ andT ₃:K → E be nonexpansive mappings with nonempty common fixed points set. Letα _n ,β _n ,γ _n ,α _n ^′ ,β _n ^′ ,γ _n ^′ ,α _n ^′′ ,β _n ^′′ andγ _n ^′′ be real sequences in [0, 1] such thatα _n +β _n +γ _n =α _n ^′ +β _n ^′ +γ _n ^′ =α _n ^′′ +β _n ^′′ +γ _n ^′′ = 1, starting from arbitraryx ₁ ∈ K, define the sequencex _n by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_1 x_n + \beta ''_n x_n + \gamma ''_n w_n ) \hfill \\ y_n = P(\alpha _n^\prime T_2 z_n + \beta _n^\prime x_n + \gamma _n^\prime v_n ) \hfill \\ x_{n + 1} = P(\alpha _n T_3 y_n + \beta _n x_n + \gamma _n u_n ) \hfill \\ \end{gathered} \right.$$ with the restrictions $\sum\limits_{n = 1}^\infty {\gamma _n }< \infty , \sum\limits_{n = 1}^\infty \gamma _n^\prime< \infty ,\sum\limits_{n = 1}^\infty {\gamma ''_n }< \infty $ . (i) If the dual E* ofE has the Kadec-Klee property, then weak convergence of ax _n to somex* ∈ F(T ₁) ∩F(T ₂) ∩ (T ₃) is proved; (ii) IfT ₁,T₂ andT ₃ satisfy condition (A′), then strong convergence ofx _n to some x* ∈F(T ₁) ∩F(T ₂) ∩ (T ₃) is obtained.     

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

Covers of the projective line and the moduli space of quadratic differentials

Consider the Hurwitz space parameterizing covers of ${\mathbb{P}^1}$ branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmüller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmüller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.     

Some extremal properties of positive trigonometric polynomials

For n=8 an upper bound is given for the functional $$V_n = \mathop {\inf }\limits_{t_n } \frac{{\alpha _1 + \alpha _2 + \cdots + \alpha _n }}{{\left( {\sqrt {\alpha _1 } - \sqrt {\alpha _0 } } \right)^2 }}$$ , which is defined on the class of even, nonnegative, trigonometric polynomials \(t_n (\phi ) = \sum\nolimits_{k = 0}^n {\alpha _k } cos k\phi \) , such that α _k ? 0 (k=0, ...,n) α₁>α₀ :V _s ? 34.54461566.     

Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

Suppose thatX ₁,X ₂, ...,X _n , ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letc _n be a maximum order of consistency. We consider a solution \(\hat \theta _n \) of the discretized likelihood equation $$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$ wherea _n (θ,r) is chosen so that \(\hat \theta _n \) is asymptotically median unbiased (AMU). Then the solution \(\hat \theta _n \) is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.     

О стРУктУРЕ цЕлОИ ФУН кцИИ, ОБРАЩАУЩЕИсь В НУль Н А жАДАННОИ пОслЕДОВАтЕльНОстИ тОЧЕк

The following result is proved. Theorem.Let λ _n ,0<λ _n ↑∞, be a sequence of positive numbers with finite density $$\sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ and let a compact set K has the following property: it intersects the real axis along the interval [a, b], where a is the very left point of K, B is the very right point of K; furthermore, K intersects every vertical straight line Re z=α, a≤α≤b, along an interval. If 1) $$F(z) \in [1,S_{ - \pi \sigma }^{\pi \sigma } \cup K(\alpha + i\pi \sigma ) \cup K(\alpha - i\pi \sigma )], \alpha \in R;$$ 2) 2) $$F( \pm \lambda _n ) = 0, n = 1,2,...,$$ then $$F(z) = A(z)e^{\alpha z} \alpha (z),$$ where $$A(z) \in [1,K], \alpha (z) = \prod\limits_1^\pi {\left( {1 - \frac{{z^2 }}{{\lambda _n^2 }}} \right)}$$ . This result generalizes the theorem of Kaz'min [3]. Three corollaries are also proved, which generalize the theorems ofBoas [1] andPólya [6]. In the theorems of Boas and Pólya, we haveF(n)=0, ?n ε Z. In our case $$F( \pm \lambda _n ) = 0, 0< \lambda _n \uparrow \infty , \sigma = \mathop {\lim }\limits_{n \to \infty } \frac{n}{{\lambda _n }}$$ .