Definitions

A topological dynamical system is a pair $$(X, T)$$ where $$X$$ is a compact metric space and let $$T:X\to X$$ is a continuous map.

By the forward-orbit of a point $$x$$ in $$X$$ we mean the set $$O_x=\{T^nx:\ n=1, 2, 3, ...\}$$.

We say that a point $$x$$ in $$X$$ is generic if $$\bar O_x$$ (closure of $$O_x$$) is $$X$$.

We say that $$x$$ is recurrent if

$$x\in \bigcap_{n\geq 1}\overline{\{T^ix:\ i\geq n\}}$$ (Thanks to @JohnB for correcting the definition of a recurrent point).

We say that the dynamical system $$(X, T)$$ is minimal if the only non-empty $$T$$-invariant closed subset of $$X$$ is $$X$$.

Question.

It is easy to show that if $$(X, T)$$ is a minimal system then every point of $$X$$ is generic and hence every point is recurrent.

Suppose $$(X, T)$$ is such that every point is recurrent. This is not enough to ensure minimality. For one can consider the closed disc $$D^2$$ with an irrational rotation.

So I am wondering if the following is true:

Let $$(X, T)$$ be a topological dynamical system such that every point of $$X$$ is recurrent and that there is at least one generic point. Does this imply that the system is minimal? What if one assumes $$T$$ is a homeomorphism and not just a continuous map?

• Given that topological dynamical systems is included into the tag dynamical-systems, is it really necessary to introduce the new tag "topological-dynamics"? – Arctic Char Aug 29 '19 at 23:38
• The tag 'dynamical-systems' did not come to my mind. – caffeinemachine Aug 30 '19 at 8:10
• Your notion "We say that $x$ is recurrent if $x\in \bar O_x$." is not the usual one in terms of $\omega$-limit sets. – John B Aug 30 '19 at 23:15
• @JohnB You are right. My thinking was cloudy. I have edited. – caffeinemachine Aug 30 '19 at 23:50
• – Mirko Sep 2 '19 at 19:26