Cluster points and asymptotic values of \(C^1\) and planar harmonic functions (Q2467673)

In 2005, the author published a paper [Trans. Am. Math. Soc. 357, No. 8, 3133--3167 (2005; Zbl 1078.30021)] where she studied the properties of complex-valued harmonic functions in the plane. These properties are summarized in the introduction of the paper, together with some similar results obtained by other authors. In the present paper the studies are extended to the \(C^1\)-functions \(f(z)=(u(z),v(z))\) in an open set \(R\subseteq\mathbb R^2\), where \(z= x+ iy\in C\), \(f= u+ iv\). Her attention is focused on the critical set of \(f\) and on the cluster set of \(f\). The critical set of \(f\) is the set of points where the Jacobian vanishes and the cluster set of \(f\) is the set of those finite values \(w\) such that there exists a sequence \(\{z_n\}\subset\mathbb R\) such that \(z_n\to\partial R\cup\{\infty\}\) and \(f(z_n)\to w\). The main results of the paper consist in two theorems. In the first theorem there are given the conditions in which a harmonic function \(f: C\to C\) has the properties: \(\text{int\,}C(f,\infty)\neq \Phi\) and \(\text{Val}(f, w)= \infty\) in a neighborhood of \(w_0\) and in the second one there are given the conditions, in which a \(C^1\)-function on \(\mathbb R^2\) has the property that \(w_0\) is an asymptotic value of \(f\). In the above assertions there were used the following notations: 1. \(C(f,\infty)\) is the cluster set of \(f\) if \(R=\mathbb R^2\), 2. \(\text{Val}(f,w):=\#\{z\in R: f(z)=w\}\). The paper consists in the proofs of these theorems and ends with some examples illustrating the obtained results.