Abstract
We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of these points and components is called the residual Julia set, denoted by , and is defined to be the subset of those points of the Julia set, chaotic set, which do not belong to the boundary of any component of the Fatou set (stable set). The points of are called buried points and the components of are called buried components. In this paper we extend some results related with the residual Julia set of transcendental meromorphic functions to functions which are meromorphic outside a compact countable set of essential singularities. We give some conditions where .
Highlights
Let X,Y be Riemann surfaces and Df be an arbitrary non-empty open subset of X
The points of Jr f are called buried points and the components of Jr f are called buried components. This is the Residual Julia set which are in the chaotic set
In [12], McMullen defined a buried component of a rational function to be a component of the Julia set which does not meet the boundary of any component of the Fatou set
Summary
Let X,Y be Riemann surfaces (complex 1-manifolds) and Df be an arbitrary non-empty open subset of X. A Fatou component for a function in class or can be periodic, pre-periodic or wandering. [8], for functions in class , that a periodic Fatou component (of arbitrary period) is doubly or infinitely connected. In [6] the authors proved that for functions in class with a finite set of singular values there are neither wandering components nor Baker domains.
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