Комбінаторний інваріант для функції, заданої на колі

Abstract

The article deals with the case of continuous functions with a finite number of local extrema defined on a one-dimensional compact manifold without boundary. The conditions of topological equivalence of continuous functions on S 1 are investigated. A combinatorial invariant for the function f is constructed and it is determined that for the class of functions in question the complete combinatorial invariants must coincide. One of the problems of topology is the study of conditions for the topological equivalence of functions on manifolds. Determining the number of topologically nonequivalent functions on a circle reduces to combinatorics of permutations. But in nature, there are functions for which at least two local extremes correspond to one critical value. In this case, a combinatorial invariant should be introduced - a set of periodic alternating sequences. However, the number of such periodic alternating sequences is not defined, so it is advisable to construct an invariant that will allow you to specify the number of certain classes of functions on a circle and obtain estimates for the general case. The purpose of the study is to investigate the relationship between continuous functions on a circle and their combinatorial invariants to determine the topological equivalence of functions. An invariant of a function that answers the questions posed is alternating sequences, with the help of which a sequence of values of a function is constructed. In the case when the number of local extremes of the function coincides with the number of critical values, then the invariant is a snake. But not all snakes are possible, because the position of the last maximum of the snake depends on the position of the first minimum of the next snake. The article considers the general case when the number of critical values of a function is not equal to the number of local extremes and the function has m global maxima (minima). The invariant of such a function is the division of the circle into arcs, the value of the function in their local extremum, form elementary snakes. It is proved that a necessary and sufficient condition for the topological equivalence of two functions on a circle is an isomorphism of partitions, which is answered by them. Keywords: partitions, arc, function on a circle, invariant, topological equivalence, local extremum