Iurchuk I. A topological equivalence of the pseudoharmonic functions.

A topological classification of continuous functions defined on a circle with a finite number of extrema is obtained in terms of their invariant that calculates a number of topologically non equivalent functions. In one dimensional case the analogue of Arnold's hypothesis is proved. For a continuous function defined in a neighborhood of a zero that is it's local minimum (maximum) the conditions of a topological equivalence to a cone under a planar curve are obtained. For the pseudoharmonic functions defined on a disk with a finite number of extrema on a boundary the invariant is constructed. It is a finite connected graph with a strict partial order on vertices. In terms of such invariant we formulated a necessary and sufficient condition of a topological equivalence of pseudoharmonic functions. Two pseudoharmonic functions are topologically equivalent if and only if there exists an isomorphism between their invariants preserving a strict partial order on them.We are interested in an inverse problem ofthe realization of a connected finite graph with a strict partial order on vertices as a combinatorial invariant of some pseudoharmonic function.