**MSC:**- 30D05 Functional equations in the complex domain, iteration and composition of analytic functions, See also {34A20, 39-XX, 58F08, 58F23}
- 54H20 Topological dynamics, See also {28Dxx, 34C35, 58Fxx}

entire (possibly transcentental) functions. In particular, a

criterion is given assuring that a given entire semigroup is

semihyperbolic. Note that a semihyperbolic semigroup $G$ admits

holomorphic scaling, that is to say, the branches of local inverses

of functions $f \in G$ are of bounded degree and that the preimages

shrink to zero in diameter.