Hiroki SUMI
On Hausdorff dimension of Julia sets of hyperbolic rational semigroups
Abstract
We consider dynamics of semigroups generated by rational functions on the
Riemann sphere. We will show that if a semigroup of rational fuctions on the
Riemann sphere is finitely generated,\ then the hyperbolicity and the
expandingness are equivalent. If a semigroup satisfies the strong open
set condition, we can construct a $\delta $-conformal measure on the Julia set.
Also the Julia set has no interior points, and furthurmore, if the
semigroup is hyperbolic, the Hausdorff dimension of the Julia set is strictly
lower than 2.
The value $\delta $ of the dimension coincides with the unique
value that allows us to construct a $\delta $-conformal measure and the
$\delta $-Hausdorff measure of the Julia set is a finite value strictly bigger
than zero.
With the method similar to that of the construction of the Patterson-Sullivan
measures we get $\delta $-subconformal measures in more general cases and we
will show that if a finitely generated rational semigroup is expanding,
then the Hausdorff dimension of the Julia set is less than the exponent
$\delta .$
submission: March 23, 1998
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