The Theory of Cluster Sets

Abstract

The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. The cluster set of a function at a particular point is the set of limit values of the function at that point which may be either a boundary point or (in the case of a non-analytic function) an interior point of the function's domain. In topological analysis, its main application is to problems arising in the theory of functions of a complex variable, with particular reference to boundary behaviour such as the theory of prime ends under conformal mapping. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions. The authors assume a general familiarity with classical function theory but include the more specialised material required for the development of the theory of cluster sets, so making the treatment accessible to graduate students.

The Beltrami equations and prime ends

“…The boundary behavior of W 1,1 loc homeomorphic solutions and the Dirichlet problem for degenerate Beltrami equations in Jordan domains have been studied, e.g., in [17][18][19]37]. Concerning the Carathéodory's theory of prime ends for the conformal mappings, we refer the reader to [4] and [5,Ch. 9].…”

A curvilinear extension of Iversen-Tsuji’s theorem for simply connected domain

“…(1) the image of the radius te'9, 0 < / < 1, determines an accessible boundary point of G", (2) Bn(w) has a radial limit with modulus 1 at e'9, and (3) v has a radial limit at e'9, then dDw -Ew is of linear measure zero.…”

A Boundary Theorem for Tsuji Functions

“…Γ F {ιv) =~ 0 implies that every point of C is a Frostman point (i.e. a point for which CRkw, e tC *) is of capacity zero) and thus belongs to the set of Weierstrass points for which C{w, e tQ ) is total [4]. From this it follows, by the maximality theorem on cluster sets (ML Since each Stolz angle of sufficiently large aperture contains infinitely many of the points zj q , no value w 0 on the Riemann sphere can be a Fatou value of the function w. On the other hand, the set of points e iQ whose symmetrically placed Stolz angle of aperture π/4 meets only finitely many of the disks D Jq is residual, and therefore the Julia points of w form a set of first category.…”