arXiv subjects
Publications and source records attributed to A. Eremenko.
We study the limit distribution of eigenvalues of a Ruelle operator (which is also called the Thurston pushforward operator) for the dynamical system $z \mapsto z^2+c$ when $c<-2$ and tends to $-2$.
We study the spectra of certain integro-differential equations arising in applications. Under some conditions on the kernel of the integral operator, we describe the non-real part of the spectrum.
For differential equations $P(y^{(k)},y)=0,$ where $P$ is a polynomial, we prove that all meromorphic solutions having at least one pole are elliptic functions, possibly degenerate.
This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970.
We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.
We show that for a real transcendental meromorphic function f, the differential polynomial f'+f^m with m > 4 has infinitely many non-real zeros. Similar results are obtained for differential polynomials f'f^m-1. We specially investigate the case of meromorphic functions with finitely many poles. We show by examples the precision of our results. One of our main tools is the Fatou theorem from complex dynamics.
Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z : ... : z^d) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d^{th} Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example on the real line), then the function maps this circle into a circle.
Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.
A simple proof is given of the necessary and sufficient condition on a triple of positive numbers A,B,C for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles A,B,C. The same condition is necessary and sufficient for the triple A,B,C to be interior angles of a spherical triangular membrane.
If f is a real entire function and ff" has only real zeros then f belongs to the Laguerre-Polya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a conjecture of Wiman in 1914.