Example of a nonrectifiable Nevanlinna contour

Nevanlinna domains with large boundaries

“…Proof. We start with the following building block, sometimes called "Mazalov's needle", see [26,Section 2]. For every sufficiently small b > 0 there exists a rational function For I ⊂ R + , E ⊂ [0, 2π) we use the notation…”

“…It means that the boundary of a Nevanlinna domain f (D) is "almost" non-rectifiable. The first example of a Jordan Nevanlinna domain with non-rectifiable boundary was constructed in [26]. The corresponding domain is also f (D), for some function f of the form (2.4) univalent in the unit disc.…”

“…Proof. We start with the following building block, sometimes called "Mazalov's needle", see [26,Section 2]. For every sufficiently small b > 0 there exists a rational function F b with simple poles Let us choose a nowhere dense compact set K of positive one-dimensional Lebesque measure on the unit circle T. We have T\K = j≥1 I j , where…”

“…Furthermore, it was shown in [2] that Nevanlinna domains may have "almost" non-rectifiable boundaries. The first example of an N -domain with non-rectifiable boundary was constructed in [26]. Finally, we mention the recent paper [27], where an example of Nevanlinna domain G such that dim H (∂G) > 1 was produced.…”

“…Constructing Nevanlinna domains with irregular (for instance nonanalytic, non-smooth, and even more irregular) boundaries is a rather difficult and delicate problem. It was considered in [23,17,2,26,27]. The detailed account of these results will be given in Section 2 below.…”