  We consider fiber-preserving complex dynamics on fiber bundles whose fibers
are Riemann spheres and whose base spaces are compact metric spaces. In this
context, without any assumption on (semi-)hyperbolicity, we show that the
fiberwise Julia sets are uniformly perfect. From this result, we show that, for
any semigroup $G$ generated by a compact family of rational maps on the Riemann
sphere of degree two or greater, the Julia set of any subsemigroup of $G$ is
uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber
bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic,
then the fiberwise Julia sets are porous, and the dynamics is weakly rigid.
Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise
maps are polynomials, then under some conditions, the fiberwise basins of
infinity are John domains. We also show that the Julia set of a rational
semigroup (a semigroup generated by rational maps on the Riemann sphere) that
is semi-hyperbolic, except at perhaps finitely many points in the Julia set,
and which satisfies the open set condition, is either porous or equal to the
closure of the open set. Furthermore, we derive an upper estimate of the
Hausdorff dimension of the Julia set.
