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 $X$.


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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.