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DCDS

We provide proof that a random backward iteration algorithm for approximating Julia sets of rational
semigroups, previously proven to work in the context of iteration of a rational function of degree two or more, extends to rational semigroups (of a certain type). We also provide some consequences of this result.

keywords:
random complex
dynamics
,
Rational semigroups
,
Julia sets
,
random iteration
,
invariant measure.
,
Markov process

DCDS

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius 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 [8].

keywords:
random complex
dynamics
,
Rational semigroups
,
random iteration
,
Julia sets
,
Markov process
,
Möbius maps
,
invariant measure.

DCDS

In [13] there is a survey of several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, along with a discussion of which of those methods can and cannot be extended to the case of semigroups. In particular that paper presents an elementary proof based on the ideas of [11] that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points. This paper serves as a brief follow up to [13] by showing that the ideas of [3] can also be used to provide an elementary proof for the semigroup case. It also touches upon some key differences between the dynamics of iteration and the dynamics of semigroups.

keywords:
random dynamics
,
Complex dynamics
,
Julia sets
,
dynamics of semigroups
,
repelling fixed points.

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