On the maximal invariant set for the map x 2 - 2 restricted to intervals

In this paper, we study the maximal invariant set of a quadratic family related to a class of unimodal maps. This family is very important and have direct application in many branches of science. In particular, we characterize when the maximal invariant of f(x) = x 2 − 2 (restricted to an interval) has a chaotic behavior.