A note on dynamics of interval extensions of interval functions

Let I = [a, b] a real compact interval and f : I → I a continuous function. Define K_c([a, b]) the class of all non empty compact subinterval of [a, b] and let f_c the natural extension of f to K_c([a, b]), that is to say, f̅_c(J) = f(J) for all J ∈ K_c([a, b]). Also, let F_c([a, b]) the class of all fuzzy-intervals with support contained in [a, b] and consider f_c the Zadeh´s extension of f to F_c([a, b]). The aim of this paper is to show some dynamical properties of f̅_c and f̂_c in relation to the Devaney´s conditions for complexity of functions and, in particular, to show the surprising difference between the dynamics on the base space X = [a, b] compared with the dynamics of the interval extension K_c([a, b]) and fuzzy-interval extension F_c([a, b]), which could be useful in the mathematical modelling of real problems.