Valence of complex-valued planar harmonic functions

The valence of a function $f$ at a point $w$ is the number of distinct, finite solutions to $f(z)=w$. Let $f$ be a complex-valued harmonic function in an open set $R\subseteq \mathbb{C}$. Let $S$ denote the critical set of $f$ and $C(f)$ the global cluster set of $f$. We show that $f(S)\cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S)\cup C(f)$ has empty interior. We also show that a component $R_0\subseteq R\mathrm{\backslash }{f}^{-1}(f(S)\cup C(f))$ is a $n_0$-fold covering of some component $\Omega_0 \subseteq $\mathbb{C}$ \backslash (f(S) \cup C(f))$\mathrm{.}If$\Omega_0$issimplyconnected,then$f$isunivalenton$R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in ${\mathbb{R}}^2$ are first stated in that form and then applied to the case of planar harmonic functions. If $f$ is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of $$\mathbb{C}$ \backslash (f(S) \cup C(f))$sharingacommonboundaryarcin$f(S) \backslash C(f)$.