The Trace Map and Integrability of the Multifunctions

In this paper we construct the quotient map (with respect to the special equivalence relation) for the restriction of the trace map F(x,y) = (xy, (x - 2)(2)) on the invariant subset of the plane that consists of unbounded curves. There is the unique curve in every point of which (with the exception of the unique point) the quotient map is two-valued and upper semicontinuous (but not lower semicontinuous). In other points this map is single-valued and continuous. We introduce also the concept of integrability for multifunctions of the above type, formulate and prove the (necessary) conditions for integrability. These conditions are based on the reduction of the above multifunctions to the multivalued skew products.