Dimensions of orbital sets in complex dynamics

TL;DR

The paper analyzes dimensions of orbital (backward) sets O_T(E) for rational maps T of degree at least 2, relating their upper box dimension to those of the initial compact set E and the Julia set J_T. Under a mild separation-type assumption (A) that E lies in a simply connected region disjoint from the postcritical set yet meeting J, the authors prove the 'expected' formula $ar{ ext{dim}}_{ ext{B}} O_T(E) = \max\{ \bar{ ext{dim}}_{ ext{B}} E, \bar{ ext{dim}}_{ ext{B}} J \}$, using inverse-branch distortions, covering estimates, and dynamical properties of T; they also show the assumption is necessary via counterexamples. They further establish that Hausdorff dimension satisfies $ ext{dim}_{ ext{H}} O_T(E) = \text{dim}_{ ext{H}} E$, and discuss when the closure of O_T(E) includes J to yield $ ext{dim}_{ ext{H}} \overline{O_T(E)} = \max\{ \text{dim}_{ ext{H}} E, \text{dim}_{ ext{H}} J \}$. The results connect the complex-dynamics setting with inhomogeneous attractor theory and Kleinian orbital sets, highlighting a unified perspective via the Sullivan dictionary and providing sharp examples that delineate the boundaries of applicability.

Abstract

Let $ E $ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between the upper box dimension of the orbital set and the upper box dimensions of the Julia set of $T$ and the initial set $ E$. Our results extend previous work on inhomogeneous iterated function systems to the setting of complex dynamical systems.

Figures (2)

• Figure 1: Orbital sets generated by standard rational maps and various compact subsets of $\mathbb{C_{\infty}}$. The maps include $T_{1}( z) =z^{2}-1,~T_{2}( z) =z^{4}+z, ~T_{3}( z) =z^{2}$ and $T_{4}(z) = z^2+z$ and the compact subsets include circles and translates of the Sierpiński triangle and the Vicsek fractal.
• Figure 2: Orbital sets generated by standard rational maps and various compact subsets of $\mathbb{C_{\infty}}$. The maps include $T_{1}(z) = z^2,~T_{2}(z)=z^{2}+3\sqrt{2}+3\sqrt{2}i,$ and $T_{3}(z) = z^4+z$ and the compact subsets are different circles and $\{3\sqrt{2}+3\sqrt{2}i\}\cup\{3\sqrt{2}+3\sqrt{2}i+\frac{1}{n}\}_{n\in \mathbb{N}}$, where $3\sqrt{2}+3\sqrt{2}i$ is a critical value of $T_{2}(z)$.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5