$X$ is a (compact) metric space, $T:X\rightarrow X$ is a continuous self-map.
Let $x\in X$, $E\subseteq X$,
$E$ is said to be $T$-invariant if $TE\subseteq E$.
$E$ is called a minimal subset of $(X,T)$, if $E$ is nonempty, closed and $T$-invariant, and has no proper subset possessing these three properties.
$x\in X$ is called a minimal point, if the closure of the orbit of $x$ is minimal.
A point $x\in X$ is said to be recurrent, if for each neighbourhoof $U$ of $x$, there exists $n\in\mathbb{N}$ such that $T^nx\in U$.
If $\{x_n\}_{n=1}^{\infty}$ is a sequence of minimal points, and $x_n\rightarrow x(\in X)$, then $x$ is a minimal point?
If $\{x_n\}_{n=1}^{\infty}$ is a sequence of recurrent points, and $x_n\rightarrow x(\in X)$, then $x$ is a recurrent point?
These things make me think for a long time, but I have no idea...
Thanks a lot.
No and no. The shift dynamical system has simple counter examples.
Let $X=\{0,1\}^\mathbb{Z}$ be the set of bi-infinite binary sequences and $T$ the shift map $(Tx)_i:=x_{i+1}$. With the product topology, $X$ is compact and metrizable, and $T$ is continuous.
Any periodic point in $X$ is recurrent and has minimal orbit closure. On the other hand, periodic points are dense in $X$. So, for any non-recurrent point $x$ that has non-minimal orbit closure, there is a sequence of periodic points converging to $x$.