Related papers: On Rational Maps with Two Critical Points

We discuss in detail the dynamics of maps $z\mapsto ae^z+be^{-z}$ for which both critical orbits are strictly preperiodic. The points which converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called…

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…

We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose…

A Belyi map $\beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$. Replacing $\mathbb{P}^1$ with an elliptic curve…

Recently the first author studied the bifurcation of critical points of families of functionals on a Hilbert space, which are parametrised by a compact and orientable manifold having a non-vanishing first integral cohomology group. We…

We study cubic rational maps that take lines to plane curves. A complete description of such cubic rational maps concludes the classification of all planarizations, i.e., maps taking lines to plane curves.

These are the substantially expanded notes of the lectures of JK at the summer school "Higher-Dimensional Geometry over Finite Fields" in G\"ottingen, June 2007. The first part gives an overview of the methods. The main new result is the…

We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New…

We describe dynamical properties of a map $\mathfrak{F}$ defined on the space of rational functions. The fixed points of $\mathfrak{F}$ are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials.

We study periodicity conditions of a rational map on $\bm{C}^d$ with $p$ invariants and show that a set of isolated periodic points and an algebraic variety of finife dimension do not exist in one map simultaneously if $p\ge d/2$. We also…

Let $K$ be a number field and $S$ a fixed finite set of places of $K$ containing all the archimedean ones. Let $R_S$ be the ring of $S$-integers of $K$. In the present paper we study the cycles for rational maps of $\mathbb{P}_1(K)$ of…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

Let R be rational map. Critical point c is called summable if series $\sum_i\frac{1}{(R^i)'(R(c))}$ is absolutely convergent. Under some topological condition on postcritical set we prove that R can not be structurally stable if summable…

In Algebraic Vision, the projective reconstruction of the position of each camera and scene point from the knowledge of many enough corresponding points in the views is called the structure from motion problem. It is known that the…

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…

We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed `Herman map' is developed.…

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy…

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…

When is a topological branched self-cover of the sphere equivalent to a rational map on CP^1? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive…