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

TL;DR

This paper establishes conditions under which pseudoharmonic functions on a disk are topologically equivalent by constructing an invariant that captures their essential features.

Contribution

It introduces a new invariant for pseudoharmonic functions on a disk and provides necessary and sufficient conditions for their topological equivalence.

Findings

Constructed an invariant containing all information about pseudoharmonic functions.

Derived necessary and sufficient conditions for topological equivalence.

Characterized pseudoharmonic functions via their critical points and boundary behavior.

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.