On Julia limiting directions in higher dimensions

Phragmén-Lindelöf 原理与拟正则映射的 Julia 极限方向

Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings

本文研究超越型拟正则映射的 Julia 极限方向集结构，利用允许 $O(\log |x|)$ 边界增长的新型 Phragmén-Lindelöf 原理，建立其与维数、伸缩率和增长阶的关系。

Fatou directions along the Julia set for endomorphisms of CP^k

Not much is known about the dynamics outside the support of the maximal entropy measure $μ$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(μ)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for…

On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at -2

Let $d(δ)$ denote the Hausdorff dimension of the Julia set of the polynomial $f_δ(z)=z^2-2+δ$. In this paper we will study the directional derivative of the function $d$ along directions landing at the parameter $0$, which corresponds to $-2$ in the case of family $p_c(z)=z^2+c$. We will consider all directions, except the one $δ\in\mathbb{R}^+$, which is inside the Mandelbrot set. We will prove asymptotic formula for the directional derivative of $d$. Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is n…

On the directional derivative of the Hausdorff dimension of quadratic polynomial Julia sets at 1/4

Let $d(\varepsilon)$ and $\mathcal D(δ)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_δ(z)=(1+δ)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(δ)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $δ$ parametrization) which is outside the Mandelbrot set and is related to the paraboli…

The distribution of directions in an affine lattice: two-point correlations and mixed moments

We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most $T$. Strömbergsson and the second author proved in [Annals of Math.~173 (2010), 1949--2033] that the distribution of gaps between the lattice directions has a limit as $T$ tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs distinctly from the gap distribution observed in a Poisson process, which is exponential. The present study shows that the limiting two-point correlation function of the projected lattice points e…

The gap distribution of directions in some Schottky groups

We prove the existence and some properties of the limiting gap distribution functions for the directions of orbits of some infinite covolume subgroups of $Isom(\mathbb{H}^2)$ in the Poincaré disk.