**MSC:**- 58F23 Holomorphic dynamics, See also {30D05}
- 30D05 Functional equations in the complex domain, iteration and composition of analytic functions, See also {34A20, 39-XX, 58F08, 58F23}

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.

for young scientists.