We think of this sequence as describing the time evolution of the point x. It will be convenient to denote the n-fold composition of f with itself by f, so f(x) is the result of n applications of f to x. In the theory of dynamical systems, we are typically interested in the long time behavior of orbits. This can depend very sensitively on small details, even in simple examples. As an illustrati...

The main goal of the present paper is to extend classical results from the measure theory and dynamical systems to the fuzzy subset setting. In this paper, the notion of  dynamic equivalence relation is introduced and then it is proved that this relation is an equivalence relation. Also, a new metric on the collection of all equivalence classes is introduced and it is proved that this metric is...