On Dynamics of hyperbolic rational semigroups
A "rational semigroup" is a semigroup generated by non-constant rational maps
on the Riemann sphere.
The Julia sets of finitely generated rational semigroups have the backward
self-similarity. Hyperbolic rational semigroups which contains an element with
the degree at least two have no wandering domains. If a finitely generated
hyperbolic rational semigroup contains an element with the degree at least two
and each Mo"bius transformation in the semigroup is loxodromic, then the limit
functions of the semigroup on the Fatou set are only constant functions that
take their values on postcritical set. When the generators of a finitely
generated hyperbolic rational semigroup are perturbed, the hyperbolicity is kept
and the Jilia set depends cotiniously on the generators of the semigroup.
Furthermore, if the finitely generated rational semigroup is hyperbolic and if
the inverse images by the generators of the Julia set are mutually disjoint,
then the Julia set moves by holomorphic motion.
Because of the backward self-similarity, if the postcritical set is included
in a Fatou component, then the Julia set has a property which is like usual
self-similarity, and moreover, if the inverse images by the generators of the
Julia set are mutually disjoint, then the Julia set is a Cantor set.
submission: March 26, 1998
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