Porosity in conformal dynamical systems

TL;DR

This paper investigates porosity properties of conformal fractals, including limit sets of graph directed Markov systems, complex continued fractions, and Julia sets, revealing conditions for porosity and mean porosity in various dynamical contexts.

Contribution

It provides new criteria and characterizations for porosity in conformal dynamical systems, extending understanding to complex continued fractions and Julia sets, and introduces concepts like upper density and box dimension.

Findings

Limit sets of GDMS are porous in large subsets and mean porous almost everywhere.

Limit sets of certain complex continued fractions are not porous almost everywhere but are mean porous.

Julia sets of non-elliptic meromorphic functions are porous at dense points and almost everywhere mean porous.

Abstract

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that, under some natural assumptions, their limit sets are porous in large (in the sense of category and dimension) subsets, and they are mean porous almost everywhere. On the other hand, we prove that if the limit set of a GDMS is not porous then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction. We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterization of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are mean porous almost everywhere. We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.