Arnold Mathematical Journal最新文献

In this paper, we continue the study, began in Chen et al. (Slices of parameter space for meromorphic maps with two asymptotic values, arXiv:1908.06028, 2019), of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values, and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In Fagella and Keen (Stable components in the parameter plane of meromorphic functions of finite type, arXiv:1702.06563, 2017) and Chen and Keen (Discrete and Continuous Dynamical Systems 39(10):5659–5681, 2019), it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here, we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle are also accessible from the shift locus.

We consider Bethe subalgebras B(C) in the Yangian ({mathrm {Y}}({mathfrak {gl}}_2)) with C regular (2times 2) matrix. We study the action of Bethe subalgebras of ({mathrm {Y}}({mathfrak {gl}}_2)) on finite-dimensional representations of ({mathrm {Y}}({mathfrak {gl}}_2)). We prove that B(C) with real diagonal C has simple spectrum on any irreducible ({mathrm {Y}}({mathfrak {gl}}_2))-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of B(C).

We consider the KZ differential equations over ({mathbb {C}}) in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field ({mathbb {F}}_p). We study the space of polynomial solutions of these differential equations over ({mathbb {F}}_p), constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over ({mathbb {F}}_p) with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over ({mathbb {F}}_p) and the hypergeometric solutions over ({mathbb {C}}).

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.