On the conditions of topological equivalence of pseudoharmonic functions defined on disk

Abstract

Let $D^2 \subset C$ be a closed two-dimensional disk and $f:D^2 \to R$ be a continuous function such that a restriction of $f$ to $\partial D^2$ is a continuous function with a finite number of local extrema and $f$ has a finite number of critical points in $\Int{D^2}$ such that each of them is saddle (i.e., in its neighborhood the local representation of $f$ is $f = Re z^n + const$, where $z=x+iy$, $n \geq 2$). This class of functions coincides with class of pseudoharmonic functions defined on $D^2$. First, we will construct an invariant of such functions which contains all information about them. Then, in terms of such invariant the necessary and sufficient conditions for pseudoharmonic functions to be topologically equivalent will be obtained.