topological dynamics – A reference to the fact that a topologically transitive action of a group on a compact metric space has a dense orbit


I need an appropriate reference to the following obvious fact:

A group action $ G $ on a compact metrable space $ K $ is topologically transitive (= the orbit of any non-empty open set is dense) if and only if it is the orbit $ Gx $ from a certain point $ x in K $ is dense in $ K $.

For a cyclic group, this characterization is proven here. I hope that some textbooks on topological dynamics should contain such a basic fact. Help me!