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Vol 77, No 6 (2022)
Iterates of holomorphic maps, fixed points, and domains of univalence
Abstract
Fixed points play an important part in the dynamics of a holomorphic map. Given a holomorphic self-map of a unit disc, all of its fixed points, with the exception of at most one of them, lie on the boundary of the disc. Furthermore, it turns out that the existence of an angular derivative and its value at a boundary fixed point affect significantly the behaviour of the map itself and its iterates. In addition, some classical problems in geometric function theory acquire new settings and statements in this context. These questions are considered in this paper. The presentation focuses on the problem of fractional iterations, domains of univalence, and the influence of the angular derivative at a boundary fixed point on the regions of values of Taylor coefficients.Bibliography: 90 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):3-68
3-68
Spectral inequality for Schrödinger's equation with multipoint potential
Abstract
Schrödinger's equation with potential that is a sum of a regular function and a finite set of point scatterers of Bethe–Peierls type is under consideration. For this equation the spectral problem with homogeneous linear boundary conditions is considered, which covers the Dirichlet, Neumann, and Robin cases. It is shown that when the energy $E$ is an eigenvalue with multiplicity $m$, it remains an eigenvalue with multiplicity at least $m-n$ after adding $n0042-1316m$ point scatterers. As a consequence, because for the zero potential all values of the energy are transmission eigenvalues with infinite multiplicity, this property also holds for $n$-point potentials, as discovered originally in a recent paper by the authors.Bibliography: 33 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):69-76
69-76
The finite-gap method and the periodic Cauchy problem for $(2+1)$-dimensional anomalous waves for the focusing Davey–Stewartson $2$ equation
Abstract
The focusing nonlinear Schrödinger equation is the simplest universal model describing the modulation instability of $(1+1)$-dimensional quasi monochromatic waves in weakly nonlinear media, and modulation instability is considered to be the main physical mechanism for the appearance of anomalous (rogue) waves in nature. By analogy with the recently developed analytic theory of periodic anomalous waves of the focusing nonlinear Schrödinger equation, in this paper we extend these results to a $(2+1)$-dimensional context, concentrating on the focusing Davey–Stewartson $2$ equation, an integrable $(2+1)$-dimensional generalization of the focusing nonlinear Schrödinger equation. More precisely, we use the finite gap theory to solve, to the leading order, the doubly periodic Cauchy problem for the focusing Davey–Stewartson $2$ equation, for small initial perturbations of the unstable background solution, which we call the doubly periodic Cauchy problem for anomalous waves. As in the case of the nonlinear Schrödinger equation, we show that, to the leading order, the solution of this Cauchy problem is expressed in terms of elementary functions of the initial data.Bibliography: 86 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):77-108
77-108
Geometry of quasiperiodic functions on the plane
Abstract
A review of the most recent results obtained in the Novikov problem of the description of the geometry of the level curves of quasiperiodic functions in the plane is presented. Most of the paper is devoted to the results obtained for functions with three quasiperiods, which play a very important role in the theory of transport phenomena in metals. In that part, along with previously known results, a number of new results are presented that refine significantly the general description of the picture arising. New statements are also presented for functions with more than three quasiperiods, which open approaches to further investigations of the Novikov problem in the most general formulation. The role of the Novikov problem in various fields of mathematical and theoretical physics is discussed.Bibliography: 60 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):109-136
109-136
On the integrability of the equations of dynamics in a non-potential force field
Abstract
A range of issues related to the integration of the equations of motion of mechanical systems in non-potential force fields (often called circulatory systems) are discussed. The approach to integration is based on the Euler–Jacobi–Lie theorem: for exact integration of a system with $n$ degrees of freedom it is necessary to have $2n-2$ additional first integrals and symmetry fields (taking the conservation of the phase volume into account) which are in certain natural relations to one another. The cases of motion in non-potential force fields that are integrable by separation of variables are specified. Geometric properties of systems with non-Noether symmetry fields are discussed. Examples of the existence of irreducible polynomial integrals of the third degree in the momentum are given. The problem of conditions for the existence of single-valued polynomial integrals of circulatory systems with two degrees of freedom and toric configuration spaces is considered. It is shown that in a typical case the equations of motion do not admit non-constant polynomial integrals.Bibliography: 32 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):137-158
137-158
Trace formula for the magnetic Laplacian at zero energy level
Abstract
The paper is devoted to the trace formula for the magnetic Laplacian associated with a magnetic system on a compact manifold. This formula is a natural generalization of Gutzwiller's semiclassical trace formula and reduces to it in the case when the magnetic field form is exact. It differs slightly from the Guillemin–Uribe trace formula considered in a previous paper of the author and Taimanov. Moreover, in contrast to that paper, the focus is on the trace formula at the zero energy level, which is a critical energy level. An overview of the main notions and results related to the trace formula at the zero energy level is presented, various approaches to its proof are described, and concrete examples of its computation are given. In addition, a brief review of Gutzwiller's trace formula for regular and critical energy levels is presented.Bibliography: 88 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(6):159-202
159-202
Asymptotic properties of Hermite–Pade polynomials and Katz points
Uspekhi Matematicheskikh Nauk. 2022;77(6):203-204
203-204
Dolzhenko's inequality for $n$-valent functions: from smooth to fractal boundaries
Uspekhi Matematicheskikh Nauk. 2022;77(6):205-206
205-206
On the Davis–Monroe problem
Uspekhi Matematicheskikh Nauk. 2022;77(6):207-208
207-208
Iskander Asanovich Taimanov (on his 60th birthday)
Uspekhi Matematicheskikh Nauk. 2022;77(6):209-218
209-218