Compact Sets in Petals and their Backward Orbits under Semigroups of Holomorphic Functions

TL;DR

This paper studies the backward orbits of compact sets in the unit disk under semigroups of holomorphic functions, analyzing their geometric and potential theoretic properties such as hyperbolic area, diameter, harmonic measure, and capacity.

Contribution

It provides new conditions for the existence of backward orbits and examines their geometric and potential theoretic characteristics in detail.

Findings

Conditions for backward orbit existence are established.

Asymptotic behavior of hyperbolic area and diameter is characterized.

Harmonic measure and capacity of the backward orbit are analyzed.

Abstract

Let $(\phi_t)_{t \geq 0}$ be a semigroup of holomorphic functions in the unit disk $\mathbb{D}$ and $K$ a compact subset of $\mathbb{D}$. We investigate the conditions under which the backward orbit of $K$ under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of $K$ are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that $K$ forms with the unit disk.