Existence of a dense orbit and topological transitivity: When are they equivalent?

Topological transitivity and existence of a dense orbit are two notions which play an important role in every definition of chaos [2], [4]. Unfortunately, in the literature there are often misunderstandings (or even false statements) about their relationship. In this note we show that generally neither condition implies the other and then we give two propositions (Propositions1 and 2) for implication in either direction.