Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in the orbit $G(x_0)$ such that $x_k\to x$ as $k$ tends to infinity. Let $L$ be the set of limit points.

Question: is $L$ always closed?

For a fixed base point $x$, the accumulation set $L_x$ of $G(x)$ is closed. The limit set $L$ can be defined as $$L=\bigcup_{x\in X}L_{x}.$$ This is a union of infinitely many closed sets, therefore not necessarily closed. I would like to see an example of limit set that is not closed, or an argument showing that $L$ is always closed. Further assumptions on $G$ and $X$ are welcome, as long as you feel comfortable.

For example: consider Kleinian group acting on the unit disk, then the limit set $L$ is on the boundary and closed. However, this example is quite special, because the orbit of any base point has the same accumulation set.

~~Another example: I believe that in the case where $G$ acts as a group of isometries, $L$ is closed.~~ This example is wrong. Thanks to Yves.