Hiroki Sumi
Dynamics of Sub-hyperbolic and Semi-hyperbolic Rational Semigroups

MSC:

58F23 Holomorphic dynamics

30D05 Functional equations in the complex domain, iteration and composition of analytic functions

Abstract: We define sub-hyperbolic and semi-hyperbolic semigroups
of rational functions on the Riemann sphere and will
show no wandering domain theorems. In particular,
if the semigroup is finitely generated, then there
exists an attractor in the Fatou set. Using that,
we investigate some properties of forward and backward
dynamics. In general, the Julia set of any semigroup
may have no empty interior points. Here, we will study
when the area of the Julia set is
equal to zero. Considering Poincare series of the semigroup,
we estimate the Hausdorff dimension of the Julia set from
above.
Keywords: rational semigroup, semi-hyperbolic, attractor, Hausdorff dimension of Julia sets
Notes: This research is partially supported by JSPS fellowship
for young scientists.