Regularly branched coverings and an application to Blaschke products with certain boundary characteristics
Ryan, Frank Beall
MacLane, G. R.
Master of Arts
The object of this thesis is two-fold: first, to treat the general existence theorems for universal (simply connected) regularly branched covering surfaces; second, to show by example how the utilization of such covering surfaces yields some interesting results in the theory of functions. In particular we shall find that there exists a Blaschke product f(z), defined in lzl < 1, which assumes as a radial limit any given value of modulus one on a set of radii having locally the power of the continuum, whose endpoints form a dense set on lzl = 1 having linear measure zero. Moreover the set of radii on which f(z) does not possess a radial limit also has locally the power of the continuum. A generalization of this example shows that, given an arbitrary perfect set E on lwl = 1, there exists a Blaschke product f(z) defined in lzl < 1 with the following properties: f(z) assumes a given value a a E as a radial limit on a set of radii having the power of the continuum, while a given value b a lwl = 1 - E is the radial limit of f(z) on a countable set of radii.