C^2 上多项式斜积映射动力学的严格数值研究

C^2 的 Axiom A 多项式斜积及其后临界集

Axiom A polynomial skew products of C^2 and their postcritical sets

本文研究 C^2 上多项式斜积的 Axiom A 性质，证明其临界轨道要么逃逸到无穷远、要么累积于吸引集，并构造新例子展示多样的后临界行为。

多项式斜积在无穷附近的 Newton 多边形与 Böttcher 坐标

Newton polygons and Böttcher coordinates near infinity for polynomial skew products

本文研究次数至少为2的多项式斜积，利用 $p$ 的次数与 $q$ 的 Newton 多边形构造无穷附近的不变区域，并证明其在该区域上共轭于单项式映射。

Non-wandering Fatou components for strongly attracting polynomial skew products

We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant attracting fiber, under the assumption that the multiplier $ λ$ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.

The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts

The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schrödinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schrödinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.

吸引型多项式斜积的游荡域问题

The wandering domain problem for attracting polynomial skew products

本文研究具有吸引型不变线的多项式斜积的游荡域问题，证明单临界条件下其吸引盆中的Fatou分量均为一维分量的扩张，从而不存在游荡Fatou分量。

Non-Smooth Bifurcations of Uniformly Hyperbolic Invariant Manifolds in Skew Product Systems: Rigorous Results

In this paper we study the anti-integrable limit scenario of skew-product systems. We consider a generalization of such systems based on the Frenkel-Kontorova model, and prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori for more general twist maps. We also prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor …

A polynomial skew-product with a wandering Fatou-disk

Little is known about the existence of wandering Fatou components for rational maps in two complex variables. In 2003 Lilov proved the non-existence of wandering Fatou components for polynomial skew-products in the neighborhood of an invariant super-attracting fiber. In fact Lilov proved a stronger result, namely that the forward orbit of any vertical disk must intersect a fattened Fatou component of the invariant fiber. Naturally the next class of maps to study are polynomial skew-products with an invariant attracting (but not super-attracting) fiber. Here we show that Lilov's stronger result…

Non-uniform hyperbolicity in polynomial skew products

Let $f:\mathbb{C}^2\to \mathbb{C}^2$ be a polynomial skew product which leaves invariant an attracting vertical line $ L $. Assume moreover $f$ restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted to $L$ satisfies one of the following conditions: 1. $f|_L$ satisfies Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov exponent at every critical value point lying in the Julia set of $f|_L$ exist and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of $L$ coincides with the uni…

Hyperbolicity and bifurcations in holomorphic families of polynomial skew products

We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …

特殊正则多项式斜积

Special regular polynomial skew products

本文研究特殊正则多项式斜积的定义与性质，证明其与仿射自映射及乘子域的关系，扩展了单变量多项式情形。