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 $\{y\in {\mathbb R} : f^n(y) = x$ for some $n$ in ${\mathbb N}\}$ is dense in ${\mathbb R}$.