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### Open Access Journals

DCDS

We estimate the Bowen parameters and the Hausdorff dimensions of the
Julia sets of expanding finitely generated rational semigroups.
We show that the Bowen parameter is larger than or equal to the ratio of the
entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with
respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if
the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation.
Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that
the Bowen parameter is strictly larger than $2$.

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

We consider the dynamics of semi-hyperbolic semigroups generated by
finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds
it is proved that there exists a geometric measure on the Julia set with exponent
$h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff
and packing measures are finite and positive on the Julia set and are mutually equivalent
with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal
dimensions, Hausdorff, packing and box counting are equal. It is also proved that for
the canonically associated skew-product map there exists a unique $h$-conformal measure.
Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely
continuous invariant (under the skew-product map) measure. In fact these two measures are
equivalent, and the invariant measure is metrically exact, hence ergodic.

DCDS

We investigate the dynamics of
semigroups generated by a family of polynomial maps on the Riemann sphere
such that the postcritical set in the complex plane is bounded.
The Julia set of such a semigroup may not be connected in general.
We show that for such a
polynomial semigroup,
if $A$ and $B$ are two connected components of the Julia set,
then one of $A$ and $B$ surrounds the other. From this,
it is shown that each connected component of the Fatou set is
either simply or doubly connected.
Moreover, we show that the Julia set of such a semigroup
is uniformly perfect.
An upper estimate of the cardinality of the set of all
connected components of the Julia set of such a semigroup is given.
By using this, we give
a criterion for the Julia set
to be connected.
Moreover, we show that for any $n\in N \cup \{ \aleph _{0}\} ,$
there exists a finitely generated polynomial semigroup with bounded
planar postcritical set
such that the cardinality of the set of all connected components of the
Julia set is equal to $n.$
Many new phenomena of polynomial semigroups that do not occur in the usual dynamics of polynomials
are found and systematically investigated.

## Year of publication

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