Properties of topologically transitive maps on the real line

Abstract

We prove that every topologically transitive map f on the real line must satisfy the following properties: (1) The set C of critical points is unbounded. (2)The set f(C) of critical values is also unbounded. (3)Apart from the empty set and the whole set, there can be at most one open invariant set. (4)With a single possible exception, for every element x the backward orbit {y∈R:fn(y)=x for some n in N} is dense in R.