Structural Instability of Semi-Siegel Hénon maps

Semi-continuity of Siegel disks under parabolic implosion

We transfer the lower semi-continuity of Siegel disks with fixed Brjuno type rotation numbers to geometric limits. Here, we restrict to Lavaurs maps associated to quadratic polynomials with a parabolic fixed point.

Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Universality and Non-Rigidity

It was recently shown by Gaidashev and Yampolsky that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal, and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalization in the Hénon family considered by de Carvalho, Lyubich and Martens. As an application of our result, we prove that the boundary of the g…

Renormalization in the Golden-Mean Semi-Siegel Hénon Family: Non-Quasisymmetry

For quadratic polynomials of one complex variable, the boundary of the golden-mean Siegel disk must be a quasicircle. We show that the analogous statement is not true for quadratic Hénon maps of two complex variables.

An entire transcendental family with a persistent Siegel disc

We study the class of entire transcendental maps of finite order with one critical point and one asymptotic value, which has exactly one finite pre-image, and having a persistent Siegel disc. After normalisation this is a one parameter family $f_a$ with $a\in\mathbb{C}^*$ which includes the semi-standard map $λze^z$ at $a=1$, approaches the exponential map when $a\to0$ and a quadratic polynomial when $a\to\infty$. We investigate the stable components of the parameter plane (capture components and semi-hyperbolic components) and also some topological properties of the Siegel disc in terms of th…

混沌动力学与零分布：在Yitang Zhang的Landau-塞格尔零定理中的意义与应用

Chaotic Dynamics and Zero Distribution: Implications and Applications in Control Theory for Yitang Zhang's Landau Siegel Zero Theorem

本研究通过分析狄利克雷L函数的混沌动力学，揭示了零点分布与量子混沌的联系，并探讨了其在控制理论中的应用潜力。

关于高维积分性和乘法依赖在半群代数动力学中的研究

On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics

本文研究高维半群轨道中点的乘法依赖性，证明了半群轨道中积分点非密集性导致轨道中乘法依赖性稀疏，扩展了Bérczes等人和Northcott、Siegel定理的结果。