Arnold Mathematical Journal最新文献

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal (S^1)-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number (frac{1}{2}dim X-dim T) is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that ({mathbb {H}}P^2/T^3cong S^5) and (S^6/T^2cong S^4), for the homogeneous spaces ({mathbb {H}}P^2={{,mathrm{Sp},}}(3)/({{,mathrm{Sp},}}(2)times {{,mathrm{Sp},}}(1))) and (S^6=G_2/{{,mathrm{SU},}}(3)). Here, the maximal tori of the corresponding Lie groups ({{,mathrm{Sp},}}(3)) and (G_2) act on the homogeneous spaces from the left. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of eight-dimensional manifolds with the action of (T^3). This class generalizes ({mathbb {H}}P^2). We prove that their orbit spaces are homeomorphic to (S^5) as well. We link this result to Kuiper–Massey theorem and its generalizations studied by Arnold.

It has been shown by Hawkins and Koss that over any given lattice, the Weierstrass (wp ) function does not exhibit cycles of Herman rings. We show that, regardless of the lattice, any elliptic function of order two cannot have cycles of Herman rings. Through quasiconformal surgery, we obtain the existence of elliptic functions of order at least three with an invariant Herman ring. Finally, if an elliptic function has order (oge 2), then we show there can be at most (o-2) invariant Herman rings.

Suppose that f is a transcendental entire function, (V subsetneq {mathbb {C}}) is a simply connected domain, and U is a connected component of (f^{-1}(V)). Using Riemann maps, we associate the map (f :U rightarrow V) to an inner function (g :{mathbb {D}}rightarrow {mathbb {D}}). It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

In translation surfaces of finite area (corresponding to holomorphic differentials), directions of saddle connections are dense in the unit circle. On the contrary, saddle connections are fewer in translation surfaces with poles (corresponding to meromorphic differentials). The Cantor–Bendixson rank of their set of directions is a measure of descriptive set-theoretic complexity. Drawing on a previous work of David Aulicino, we prove a sharp upper bound that depends only on the genus of the underlying topological surface. The proof uses a new geometric lemma stating that in a sequence of three nested invariant subsurfaces the genus of the third one is always bigger than the genus of the first one.

We describe the topology of stable simple multisingularities of Lagrangian and Legendrian maps. In particular, the tables of adjacency indices of monosingularities to multisingularities are given for generic caustics and wave fronts in spaces of small dimensions. The paper is an extended version of the author’s talk in the International Conference “Contemporary mathematics” in honor of the 80th birthday of V. I. Arnold (Moscow, Russia, 2017).

Point-vortex dynamics describe idealized, non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. Integrability results for few point-vortices on various domains is a vivid topic, with many results and techniques scattered in the literature. Here, we give a unified framework for proving integrability results for (N=2), 3, or 4 point-vortices (and also more general Hamiltonian systems), based on symplectic reduction theory. The approach works on any two-dimensional manifold with a symmetry group; we illustrate it on the sphere, the plane, the hyperbolic plane, and the flat torus. A systematic study of integrability is prompted by advances in two-dimensional turbulence, bridging the long-time behaviour of 2D Euler equations with questions of point-vortex integrability. A gallery of solutions is given in the appendix.

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by (delta _{k,n}) the average number of projective k-planes in ({mathbb {R}}mathrm {P}^n) that intersect ((k+1)(n-k)) many random, independent and uniformly distributed linear projective subspaces of dimension (n-k-1). They called (delta _{k,n}) the expected degree of the real Grassmannian ({mathbb {G}}(k,n)) and, in the case (k=1), they proved that:

Here we generalize this result and prove that for every fixed integer (k>0) and as (nrightarrow infty ), we have

where (a_k) and (b_k) are some (explicit) constants, and (a_k) involves an interesting integral over the space of polynomials that have all real roots. For instance:

Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for (delta _{1,n}) involving a one-dimensional integral of certain combination of Elliptic functions.