Rich STANKEWITZ, Toshiyuki SUGAWA and Hiroki SUMI

*Some counterexamples in dynamics of rational semigroups*

### Abstract

We give an example of two rational functions with non-equal Julia
sets that generate a rational semigroup whose completely invariant
Julia set is a closed line segment. We also give an example of
polynomials with unequal Julia sets that generate a non nearly
Abelian polynomial semigroup with the property that the Julia set
of one generator is equal to the Julia set of the semigroup. These
examples show that certain conjectures in the field of dynamics of
rational semigroups do not hold as stated and therefore require
modification.

submission: 22 October 2001

revision: 29 October 2001

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