Dynamics of meromorphic functions with direct or logarithmic singularities

Abstract

Abstract. Let f be a transcendental meromorphic function and denote by J(f ) the Julia set and by I(f ) the escaping set. We show that if f has a direct singularity over infinity, then I(f ) has an unbounded component and I(f ) ∩ J(f ) contains continua. Moreover, under this hypothesis I(f ) ∩ J(f ) has an unbounded component if and only if f has no Baker wandering domain. If f has a logarithmic singularity over infinity, then the upper box dimension of I(f ) ∩ J(f ) is 2 and the Hausdorff dimension of J(f ) is strictly greater than 1. The above theorems are deduced from more general results concerning functions which have "direct or logarithmic tracts", but which need not be meromorphic in the plane. These results are obtained by using a generalization of Wiman-Valiron theory. This method is also applied to complex differential equations.

On the packing dimension of the Julia set and the escaping set of an entire function

“…Theorem B has been extended to meromorphic functions with finitely many poles [31] and in fact to meromorphic functions with a logarithmic tract [10,Theorem 1.4]. It is conceivable that our result admits similar extensions.…”

“…Considerable attention has been paid to the dimensions of Julia sets of entire functions; see [36] for a survey, as well as [3,4,8,9,10,27,28,34] for some recent results not covered there. Many results in this area are concerned with the Eremenko-Lyubich class B consisting of all transcendental entire functions for which the set of critical and finite asymptotic values is bounded.…”

A non-power series approach to Wiman-Valiron type theorems

“…In this note we show that the methods in [2] and [1] can also be modified to give a non-power series proof and improvement to the main result in [8] as well as a more general version of the main result in [4]. Finally by assuming that f has finite order we improve a general result of Bergweiller on the size of the set where (1.1) holds.…”

“…Thus some restriction on the functions akin to admissible is a necessary one. After the proof of Theorem 2.1 we will show that the subharmonic functions considered in [2] and [1] are automatically admissible and as such this theorem can be considered a generalization of Theorem 2.2 in [2] or Theorem 1.1 in [1].…”

“…Using only certain convexity properties of the maximum of subharmonic functions and standard harmonic measure estimates, Bergweiller, Rippon and Stallard [1,2] were able to extend (1.1) to certain classes of meromorphic functions. Modifying their methods, Langley and Rossi [11] obtained similar results for an analagous class of meromorphic functions in the unit disc.…”