on the almost sure spiraling of geodesics in negatively curved manifolds given a negatively curved geodesic metric space m we study the almost sure asymptotic penetration behavior of locally geodesic lines of m into small neighborhoods of points of closed geodesics and of other compact locally convex subsets of m we prove khintchine type and logarithm law type results for the spiraling of geodesic lines around these objets as a consequence in the tree setting we obtain diophantine approximation results of elements of non archimedian local fields by quadratic irrational ones 1

on douady earle extension and the contractibility of the vmo teichmuller space abstract douady earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self map of beltrami coefficients in this note we show that is also continuous under a stronger topology on those beltrami coefficients which induce vanishing carleson measures and thus project to the vmo teichmuller space an immediate consequence is that the vmo teichmuller space is contractible

a note on qp domains abstract in this paper we first give a new characterization for a q p 0 domain 0 p 1 in terms of a quasiconformal extension of a univalent function as an application using this characterization we obtain two more characterizations of such domains by means of some kernel functions introduced recently by hu and shen 2012 9

weil petersson teichmuller space iii dependence of riemann mappings for weil petersson curves the classical riemann mapping theorem implies that there exists a so called riemann mapping which takes the upper half plane onto the left domain bounded by a jordan curve in the extended complex plane the primary purpose of the paper is to study the basic problem how does a riemann mapping depend on the corresponding jordan curve we are mainly concerned with those jordan curves in the weil petersson class namely the corresponding riemann mappings can be quasiconformally extended to the whole plane with beltrami coefficients being square integrable under the poincare metric after giving a geometric characterization of a weil petersson curve we endow the space of all normalized weil petersson curves with a new real hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex hilbert manifold structure which implies that an appropriately chosen riemann mapping depends continuously on a weil petersson curve and vice versa this can be considered as the first result about the continuous dependence of riemann mappings on non smooth jordan curves

integrable teichmuller space abstract this paper deals with the p integrable teichmuller space and gives an intrinsic characterization of a p integrable asymptotic affine homeomorphism for p 2 more precisely it is proved that a sense preserving homeomorphism h on the unit circle is p integrable asymptotic affine namely h can be extended to a quasiconformal mapping to the unit disk whose beltrami coefficient is p integrable in the poincare metric if and only if h is absolutely continuous such that log h belongs to the besov class b p s 1

on qk teichmuller spaces abstract recently wulan and ye 36 introduced the q k teichmuller spaces and proved some theorems about the theory of q k teichmuller spaces in this note we continue to consider this theory and obtain some results on the q k teichmuller spaces in particular we establish some new characterizations of the little q k teichmuller spaces

jarnik type inequalities it is well known due to jarnik 19 that the set badr of badly approximable numbers is of hausdorff dimension one if badr c denotes the subset of x badr for which the approximation constant c x c then jarnik was in fact more precise and gave nontrivial lower and upper bounds of the hausdorff dimension of badr c in terms of the parameter c 0 our aim is to determine simple conditions on a framework which allow to extend jarnik s inequality to further examples among the applications we discuss the set badrn of badly approximable vectors in r with weights r and the set of geodesics in the hyperbolic space h which avoid a suitable collection of convex sets

analytic solutions of the 2 1 dimensional nonlinear evolution equations using the sine cosine method in this paper we establish exact solutions for 2 1 dimensional nonlinear evolution equations the sine cosine method is used to construct exact periodic and soliton solutions of 2 1 dimensional nonlinear evolution equations many new families of exact traveling wave solutions of the 2 1 dimensional boussinesq breaking soliton and bkp equations are successfully obtained these solutions may be important of significance for the explanation of some practical physical problems it is shown that the sine cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics

a note on the bmo teichmuller space abstract astala zinsmeister 2 introduced a new topology on the set of all strongly quasisymmetric homeomorphisms of the unit circle denoted by sqs s 1 by means of bmo norm under this new topology we prove that sqs s 1 is a partial topological group and the characteristic topological subgroup of sqs s 1 consists of strongly symmetric homeomorphisms

solutions of the generalized kdv equation with time dependent damping and dispersion solitary wave solutions are obtained for the generalized korteweg de vries gkdv equation with time dependent damping and dispersion by using the tanh coth method the exp function method and the modified sine cosine method these methods are useful and efficient and a variety of solitary wave solutions are obtained that possess variable coefficients

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

exact solutions for the nonlinear schrodinger equation with variable coefficients using the generalized extended tanh function the sine cosine and the exp function methods abstract in this article we find the exact traveling wave solutions of the generalized nonlinear schrodinger gnls equation with variable coefficients using three methods via the generalized extended tanh function method the sine cosine method and the exp function method the main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation

symbolic dynamics for geodesic floes the subject of symbolic dynamics is of central importance in the modern theory of dynamical systems the use of symbolic sequences to study dynamical properties of geodesies originates in the work of koebe 21 22 and morse 24 25 and is already foreshadowed in hadamard 15 and jordan 19 the method of koebe and of morse is to code a geodesic on a surface m of negative curvature by recording the order in which it traverses a given set of labelled curves on m the treatment of morse allows variable curvature but assumes at least two boundary components whereas koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface this last case is handled by recording crossings of a fixed pants decomposition anticipating the thurston parameterization of simple curves as described in 11 both koebe and morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense transitive geodesies morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion in modern terminology a minimal nowhere dense set later 26 morse treated the special closed surfaces of genus g associated to tesselations of the disc d by regular 4 gons each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin face m the set of isometries appearing as labels generate tti m thus any geodesic is coded as a doubly infinite sequence of generators of iri m the same method applies quite generally to tesselations associated to any fuchsian group we call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric the difficulty of course is to determine precisely the class of sequences which occur for a surface with boundary one obtains exactly reduced sequences in the generators in generalthe problem4s gompligated1 henge the difficulties encountered in 26 see also theorem 3 below there is another method of coding geodesies using certain boundary expansions for points at infinity in the universal cover of m for the modular surface h sl 2 2 the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the nielsen boundary expansions of 27

diophantine approximation for negatively curved manifolds let m be a geometrically finite pinched negatively curved riemannian manifold with at least one cusp inspired by the theory of diophantine approximation of a real or complex number by rational ones we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it we define a new invariant for m thehurwitz constant of m it measures how well all geodesic lines starting from the cusp are approximated by ones returning to it in the case of constant curvature we express the hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood using the cut locus of the cusp we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties we prove that the modular once punctured hyperbolic torus has the minimum hurwitz constant in its moduli space

universal teichmuller space and bmo abstract we first give some new characterizations on bmoa teichmuller space and various characterizations on vmoa teichmuller space as well in particular we prove that a quasisymmetric conformal welding h corresponds to an asymptotically smooth curve in the sense of pommerenke 1978 32 precisely when h is absolutely continuous with log h vmo we then show that these bmo teichmuller spaces have natural complex structures

bmo teichmuller spaces revisited abstract in 6 the equivalence among three definitions of bmo teichmuller spaces associated with a fuchsian group was proven using the douady earle extension operator in this paper we show that these equivalences are actually biholomorphisms in 6 it was further shown that the douady earle extension operator is continuous at the origin we improve this result by showing g teaux differentiability at this point

khinchin s theorem for approximation by integral points on quadratic varieties we prove an analogue of the khinchin s theorem for the diophantine approximation by integer vectors lying on a quadratic variety the proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group we show that in this system generic trajectories of a certain geodesic flow visit a family of shrinking subsets infinitely often

quasiconformal mappings and chord arc curves given a quasiconformal mapping p on the plane what conditions on its dilatation guarantee that p r is rectifiable and pir is locally abso lutely continuous we show in this paper that if satisfies certain quadratic carleson measure conditions with small norm then p r is a chord arc curve with small constant and p x p 0 g0 ea t dt for x e r with a e bmo having small norm conversely given any such map from r c we show that it has an extension to c with the right kind of dilatation similar results hold with 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 let p be a quasiconformal map of the plane onto itself thus p is a homeomorphism with locally integrable distributional derivatives and ps upz where u e l c il ul100 1 here we use the notations fz af azf 2 z id f fz 2 aa ida f this function u is called the complex dilatation of p the mapping theorem for q c maps states that for each u e l c il ul100 1 there is a q c map p on c with dilatation u and p is unique up to normalization a basic problem is to understand how geometric properties of p are reflected in u for example one would like to have natural conditions on u which imply that p r is rectifiable and p ir is absolutely continuous this question arises naturally when considering problems in conformal mappings and conformal welding in this paper we obtain such estimates for the mapping theorem and we also give some applications our results involve bmo aoo weights chord arc curves and carleson measures and so we first review the appropriate definitions a locally integrable function f on r lies in bmo if llfil sup lil if x fil dx received by the editors june 1 1986 1980 mathematics subject classification 1985 revision primary 30g60 secondary 42b20 30e20

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

higher bers maps and bmo teichmuller space abstract we prove that the higher bers maps induced by the higher schwarzian differential operators are holomorphic in bmo teichmuller space and their derivatives at the origin are linear bounded and surjective

solitary solutions periodic solutions and compacton like solutions using the exp function method in this paper the exp function method is used for seeking solitary solutions periodic solutions and compacton like solutions of nonlinear differential equations the combined kdv mkdv equation and the liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method

the tanh coth and the sine cosine methods for kinks solitons and periodic solutions for the pochhammer chree equations abstract in this paper the tanh coth method and the sine cosine method are used for a reliable treatment of the pochhammer chree equations which include a variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the two methods demonstrate power reliability and efficiency

spiraling spectra of geodesic lines in negatively curved manifolds given a negatively curved geodesic metric space m we study the asymptotic penetration behaviour of geodesic lines of m in small neighbourhoods of closed geodesics and of other compact convex subsets of m we define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects we prove analogs of the theorems of dirichlet hall and cusick in this context as a consequence we obtain diophantine approximation results of elements of mathbb r mathbb c or the heisenberg group by quadratic irrational ones