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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?

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A bit unfortunate this "everything worth knowing", but probably Brown is only giving a brief rough motivation at that point (his book is very good!).

It also works, say, without changes, for an ergodic measure or, in case you are not so happy with this example, with any dynamics that is uniquely ergodic (note that unique ergodicity is invariant under topological conjugacy, as is minimality). Moreover, many dynamics are not minimal but still have plenty dense orbits (example: any transitive hyperbolic toral automorphism has uncountably many dense orbits).

Better something along these lines:

Agreeing that the most basic objects of a dynamics are the smallest possible invariant objects, namely orbits, when an orbit is dense one could hope to be able to obtain all the information from the topological point of view from that single orbit.

Well, I wrote "hope". For example, to be dense doesn't say anything about the density of the orbit...

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