A topological dynamical system $(X,f)$ is called minimal if whenever there is a closed $Y \subseteq X $ s.t. $f(Y)\subseteq Y$ then either $Y = \emptyset$ or $Y=X$.
James R. Brown writes in his book Ergodic Theory and Topological Dynamics (p. 46):
The essential idea of minimality is that everything worth knowing about the system can be determined from the present and future, or from the past, present, and future, situation of a single point under the action of $f$.
What is that "everything worth knowing"? How could it be determined? Could someone elaborate on the quote or share his own understanding of minimal dynamical systems?