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This paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set. [ABSTRACT FROM AUTHOR]

We investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect. [ABSTRACT FROM PUBLISHER]

It is shown that the dnth Chebyshev polynomials on the Julia set JP , and on the equipotential curve ΓP (R), of the polynomial P(z) = zd - c, are identical and exactly equal to the nth iteration of P(z) itself. As an application, the capacity of the Julia set Jp is deduced, a result that was first obtained by Brolin. [ABSTRACT FROM AUTHOR]