A multifunctional extension of function spaces: chaotic systems are maximally ill-posed

Abstract

Abstract - We define a notion of graphical convergence of a net of functions between topological spaces X and Y . This convergence, which differs from that of Kuratowsky by being basically function theoretic rather than set theoretic, is used to define a space of multifunctions Multi(X, Y ) as a natural extension of function spaces, in as far as functional relations are expected to be dense in the space of all relations between two sets. This convergence admits a new definition of continuity of multifunctions that is defined in Section 3. Multi(X, Y ) appears to be a natural space for the evolution of discrete dynamical systems towards chaos, and Sections 4 and 5 explore this possibility based on the fundamental ill-posedness of the problem.