[Paper Review] Examples of non-autonomous basins of attraction-II

Sayani Bera|arXiv (Cornell University)|Jan 19, 2018

Advanced Differential Equations and Dynamical Systems9 references3 citations

TL;DR

This paper extends the construction of non-autonomous basins of attraction in complex space, proving the existence of countably many disjoint Short $\mathbb{C}^k$ domains in $\mathbb{C}^k$ for $k \geq 2$, and constructing a Short $\mathbb{C}^k$ that is not Runge when $k \geq 4$. It introduces a new perturbation method under uniform upper-bound conditions on automorphisms, enabling biholomorphic equivalence between basins, and provides a constructive proof of a Short $\mathbb{C}^k$ with boundary of Hausdorff dimension strictly greater than $2k-1$. The key contribution is a robust framework for generating diverse, complex Short $\mathbb{C}^k$ domains with specific geometric and dynamical properties.

ABSTRACT

The aim of this article is to enlarge the list of examples of non-autonomous basins of attraction from our previous paper and at the same time explore some other properties that they satisfy. For instance, we show the existence of countably many disjoint Short $\mathbb{C}^k$'s in $\mathbb{C}^k.$ We also construct a Short $\mathbb{C}^k$ which is not Runge and exhibit yet another example whose boundary has Hausdorff dimension $2k.$

Motivation & Objective

• To extend the list of known examples of non-autonomous basins of attraction, particularly Short $\mathbb{C}^k$ domains, beyond previous constructions.
• To demonstrate the existence of countably many pairwise disjoint Short $\mathbb{C}^k$ domains in $\mathbb{C}^k$ for $k \geq 2$.
• To construct a Short $\mathbb{C}^k$ that is not Runge, establishing a new class of non-Runge domains in higher dimensions.
• To develop a perturbation method for sequences of automorphisms satisfying the uniform upper-bound condition, ensuring biholomorphic equivalence of resulting basins.
• To provide a constructive proof of a Short $\mathbb{C}^k$ with boundary of Hausdorff dimension strictly greater than $2k-1$, using the new perturbation framework.

Proposed method

• The paper uses sequences of automorphisms $\{F_n\} \subset \mathrm{Aut}_0(\mathbb{C}^k)$ satisfying a uniform upper-bound condition at the origin, i.e., $\|F_n(z)\| < C\|z\|$ for $z$ in a fixed ball and $0 < C < 1$, to define non-autonomous basins $\Omega_{\{F_n\}}$.
• It applies a novel perturbation technique inspired by Dixon–Esterle, Glovebnik, and Stensønes, allowing small perturbations of automorphisms while preserving biholomorphic equivalence of the resulting basins.
• The construction involves inductively defining compact sets $K^i$ and perturbing automorphisms $F_n$ to ensure polynomial convexity and control over the boundary structure of the basin.
• A key component is the use of a sequence $\{a_n\}$ with $a_{n+1} < a_n^2 < 1$ and $\lim_{n\to\infty} a_n^{-2^n} = 0$, ensuring convergence and control in the dynamics of the maps.
• The method relies on the existence of a sequence $\{\delta_n\}$ with $\delta_n \to 0$ such that if $\|F_n(z) - S_n(z)\| < \delta_n$ on a fixed ball, then $\Omega_{\{F_n\}} \cong \Omega_{\{S_n\}}$, establishing stability under small perturbations.
• The proof of chaotic boundary structure uses a recursive construction of compact sets $K^i$ and perturbations that ensure the boundary has positive $2k$-dimensional measure and Hausdorff dimension exceeding $2k-1$.

Experimental results

Research questions

• RQ1 Can countably many disjoint Short $\mathbb{C}^k$ domains be constructed in $\mathbb{C}^k$ for $k \geq 2$?
• RQ2 Does there exist a Short $\mathbb{C}^k$ that is not Runge when $k \geq 4$?
• RQ3 Can the basin of attraction of a sequence of automorphisms with higher-order terms (beyond monomials) still yield a Short $\mathbb{C}^k$?
• RQ4 What conditions ensure that two non-autonomous basins of attraction are biholomorphically equivalent?
• RQ5 What is the maximal possible Hausdorff dimension of the boundary of a Short $\mathbb{C}^k$ domain?

Key findings

• The paper constructs countably many pairwise disjoint Short $\mathbb{C}^k$ domains in $\mathbb{C}^k$ for $k \geq 2$, extending the known examples beyond isolated instances.
• It exhibits a Short $\mathbb{C}^k$ domain in $\mathbb{C}^k$ for $k \geq 4$ that is not Runge, demonstrating that such domains can fail the Runge property.
• A new perturbation method is established: if $\|F_n(z) - S_n(z) less \delta_n$ on a fixed ball and $\{S_n\}$ satisfies the uniform upper-bound condition, then $\Omega_{\{F_n\}} \cong \Omega_{\{S_n\}}$, ensuring stability of the basin structure.
• The paper provides a constructive proof of a Short $\mathbb{C}^k$ with boundary of Hausdorff dimension strictly greater than $2k-1$, using recursive compact set constructions and perturbations.
• It proves that the boundary of the constructed Short $\mathbb{C}^k$ has positive $2k$-dimensional Hausdorff measure, implying the boundary is large in a geometric sense.
• The existence of a Short $\mathbb{C}^k$ with chaotic boundary is confirmed via a recursive construction where the boundary contains a dense set of compact sets with positive $2k$-dimensional measure.