A dynamical characterization of sub-Arakelian subsets

We are going to characterize those sets which can be covered by an Arakelian set in terms of dynamical properties of entire functions via similarities. Moreover, if we consider the set of universal entire functions via similarities that are bounded on such a sub-Arakelian set, then it is shown that its algebraic size is as large as possible. As a consequence, we prove that the set of universal entire functions bounded on every (straight) line is algebraically large.