The initial and terminal cluster sets of an analytic curve

TL;DR

This paper investigates the possible limit sets of an analytic curve at its endpoints, showing they can be any two continua in the extended complex plane.

Contribution

It characterizes the initial and terminal cluster sets of analytic curves, demonstrating their flexibility in forming arbitrary continua.

Findings

Initial and terminal cluster sets can be any continua in the extended complex plane.

The paper provides a construction method for such curves with prescribed limit sets.

It advances understanding of boundary behaviors of analytic curves.

Abstract

For an analytic curve $\gamma:(a,b)\rightarrow \C,$ the set of values approached by $\gamma(t),$ as $t\searrow a$ and as $t\nearrow b$ can be any two continuua of $\C\cup\{\infty\}.$