Slices of parameter space for meromorphic maps with two asymptotic values

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work by Goldberg and Keen [The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift. Invent. Math.101(2) (1990), 335–372] on degree two rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

On the section conjecture and Brauer–Severi varieties

J. Stix proved that a curve of positive genus over $$\mathbb {Q}$$ Q which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction $$R_{k/\mathbb {Q}}X$$ R k / Q X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction $${\text {cor}}_{k/\mathbb {Q}}([P])\in {\text {Br}}(\mathbb {Q})$$ cor k / Q ( [ P ] ) ∈ Br ( Q ) is non-trivial, then X satisfies the section conjecture.

Periodic points of post-critically algebraic holomorphic endomorphisms

A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$ , a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that, when $n=2$ , the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.

The μ-Basis of Improper Rational Parametric Surface and Its Application

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

The Hausdorff dimension of the Julia sets concerning generated renormalization transformation

<abstract><p>Considering a family of rational map $ {U_{mn\lambda }} $ of the renormalization transformation of the generalized diamond hierarchical Potts model, we give the asymptotic formula of the Hausdorff dimension of the Julia sets of $ {U_{mn\lambda }} $ as the parameter $ \lambda $ tends to infinity, here</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {U_{mn\lambda }} = {\left[ {\frac{{{{\left( {z + \lambda - 1} \right)}^n} + \left( {\lambda - 1} \right){{\left( {z - 1} \right)}^n}}}{{{{\left( {z + \lambda - 1} \right)}^n} - {{\left( {z - 1} \right)}^n}}}} \right]^m}, $\end{document} </tex-math></disp-formula></p> <p>where $ m \ge 2 $, $ n \ge 2 $ are two natural numbers, $ \lambda \in {{\mathbb{C}} } $.</p></abstract>

Subhyperbolic rational maps on boundaries of hyperbolic components

<p style='text-indent:20px;'>In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> still lies on <inline-formula><tex-math id="M2">\begin{document}$ \partial \mathcal{H} $\end{document}</tex-math></inline-formula>. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.</p>

JNR monopoles

We review the theory of JNR, mass $\frac{1}{2}$ hyperbolic monopoles in particular their spectral curves and rational maps. These are used to establish conditions for a spectral curve to be the spectral curve of a JNR monopole and to show that the rational map of a JNR monopole arises by scattering using results of Atiyah. We show that for JNR monopoles the holomorphic sphere has a remarkably simple form and show that this can be used to give a formula for the energy density at infinity. In conclusion, we illustrate some examples of the energy density at infinity of JNR monopoles.

Degree of Rational Maps and Specialization

One considers the behavior of the degree of a rational map under specialization of the coefficients of the defining linear system. The method rests on the classical idea of Kronecker as applied to the context of projective schemes and their specializations. For the theory to work, one is led to develop the details of rational maps and their graphs when the ground ring of coefficients is a Noetherian domain.

Reduced dynamical systems

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$ . We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.