arXiv subjects

Jianyong Qiao

Publications and source records attributed to Jianyong Qiao.

7 recordsLinked to original sources

arXiv2026

Optimal Linear Dependence on Boundary Type for Local Gromov Hyperbolicity of the Kobayashi Metric

The Gromov hyperbolicity constant of a metric space $(X,d)$ is the infimum of all $δ\ge0$ such that $(X,d)$ is $δ$-hyperbolic. For a Kobayashi hyperbolic domain $Ω\subset\C^n$ and a boundary point $p\in\partialΩ$, let $δ_{\mathrm{loc}}(Ω,p)$ denote the local Gromov hyperbolicity constant obtained by restricting the points to arbitrarily small Euclidean neighborhoods of $p$, while distances are still measured by the ambient Kobayashi distance $K_Ω$. For each even integer $M\ge2$, let $\mathfrak H(M)$ be the supremum of these constants over all complex dimensions $n\ge2$ and all domains whose boundary is smooth and convex near the distinguished point and has D'Angelo type at most $M$ at that point. We prove \[ \frac{\log 2}{2} M \le \mathfrak H(M) < 36M. \] Thus the optimal universal dependence of the local Gromov hyperbolicity constant on boundary type is linear. We also show that no analogous bound holds for the global Gromov hyperbolicity constant, even among smooth bounded strongly convex domains.

math.CV↗

arXiv2026

Arithmetic conditions for Julia sets to have positive area

In this paper, for quadratic polynomials with Cremer points or Siegel disks whose boundaries don't contain critical points, we give an arithmetic condition for their Julia sets to have positive area. Our proof relies on the Siegel Return Machinery as well as the pseudo-Siegel disk theory developed by Dudko and Lyubich.

math.DS↗

arXiv2025

Renormalization group based implicit function approach to connecting orbits

Connecting orbits are important invariant structures in the state space of nonlinear systems and various techniques are designed for their computation. However, a uniform analytic approximation of the whole orbit seems rare. Here, based on renormalization group, an implicit function scheme is designed to effectively represent connections of disparate types, where coefficients of the defining function satisfy a set of linear algebraic equations, which greatly simplifies their computation. Unknown system parameters are conveniently determined by minimizing an error function. Symmetry may be profitably utilized to reduce the computation load. Homoclinic or heteroclinic connections are found in five popular examples approximately or exactly, demonstrating the effectiveness of the new scheme.

math-ph↗

arXiv2023

On the area rate of perturbed Siegel disks

In the famous work by Buff and Chéritat constructing quadratic Julia sets with positive area, the control of the shape of perturbed Siegel disks is a key technique. To do it, Buff and Chéritat added a restrictive condition on rotation numbers. In this paper, we show that this restrictive condition can be deleted. As a consequence, more quadratic Julia sets with positive area can be constructed.

math.DS↗

arXiv2023

On quadratic Siegel disks with a class of unbounded type rotation numbers

In this paper we explore a class of quadratic polynomials having Siegel disks with unbounded type rotation numbers. We prove that any boundary point of Siegel disks of these polynomials is a Lebesgue density point of their filled-in Julia sets, which generalizes the corresponding result of McMullen for bounded type rotation numbers. As an application, this result can help us construct more quadratic Julia sets with positive area. Moreover, we also explore the canonical candidate model for quasiconformal surgery of quadratic polynomials with Siegel disks. We prove that for any irrational rotation number, any boundary point of ``Siegel disk'' of the canonical candidate model is a Lebesgue density point of its ``filled-in Julia set'', in particular the critical point $1$ is a measurable deep point of the ``filled-in Julia set''.

math.DS↗

arXiv2020

Obstacles to periodic orbits hidden at fixed point of holomorphic maps

Let $f:(\mathbb{C}^n,0)\mapsto(\mathbb{C}^n,0)$ be a germ of an $n$-dimensional holomorphic map. Assume that the origin is an isolated fixed point of each iterate of $f$. Then $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$, the sequence of the maximal number of periodic orbits of period $q$ that can be born from the fixed point zero under a small perturbation of $f$, is well defined. According to Shub-Sullivan, Chow-Mallet-Paret-Yorke and G. Y. Zhang, the linear part of the holomorphic germ $f$ determines some natural restrictions on the sequence(cf. Theorem 1.1). Later, I. Gorbovickis proves that when the linear part of $f$ is contained in a certain large class of diagonal matrices, it has no other restrictions on the sequence only when the dimension $n\leq2$ (cf. Theorem 1.3). In this paper for the general case we obtain a sufficient and necessary condition that the linear part of $f$ has no other restrictions on the sequence $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$, except the ones given by Theorem 1.1.

math.DS↗

arXiv2020

Area of Julia sets of non-renormalizable cubic polynomials

The long-standing problem of existence of nowhere dense rational Julia set with positive area has been solved by an example in quadratic polynomials by Buff and Chéritat. Since then many efforts have been devoted to finding out new classes of rational maps with nowhere dense Julia sets having positive area. So far, all known examples of this kind are renormalizable with only one exception which is a quadratic polynomial. In this paper, by developing a new approach, we prove that there exists a non-renormalizable cubic polynomial having a Julia set with positive area.

math.DS↗