Backward Iteration Algorithms for Julia sets of M\"obius Semigroups

TL;DR

This paper extends the applicability of the Random Backward Iteration algorithm to a broader class of M"obius semigroups, enabling more effective visualization of their Julia sets, including cases previously not well-understood.

Contribution

It generalizes the algorithm for M"obius semigroups, covering new scenarios where the Julia set is not a thick attractor, expanding the theoretical framework.

Findings

Extended the algorithm to new classes of M"obius semigroups

Demonstrated effectiveness in cases with non-thick attractor Julia sets

Broadened understanding of Julia set visualization techniques

Abstract

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-M\"obius element) to a class of M\"obius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick attractor in the sense given in [David Fried, Sebastian. M. Marotta, and Rich Stankewitz, Complex dynamics of M\"obius semigroups, Ergodic Theory Dynam. Systems, 32(6):1889--1929, 2012].