$\DeclareMathOperator\Sym{Sym}$I have a question about what I shall name here "transfinite dynamics" because it involves iterating a topological dynamical system $G \curvearrowright X$ through ordinals. I do not know whether this concept already has a name. If so, I would appreciate being directed to some references.

The intuition behind this construction is that the phase space $X$ will be the "board" of some game and $G$ will be the set of moves that the player is allowed to execute. I wish to understand what happens if the player is allowed to execute transfinitely many moves as long as the board "stabilizes" at limit stages.

To be more precise, let $G$ act on a Hausdorff space $X$. Extend $X$ to $\overline{X}$ by adding a point $\ast \notin X$ as an isolated point and extend the action of $G$ to $\overline{X}$ trivially. Let $\lambda$ be an ordinal and $\mathbf{g}=(g_{\alpha})_{\alpha < \lambda}$ be a sequence of elements of $G$. For each $x \in X$, define $(x_{\alpha})_{\alpha \leq \lambda}$ by transfinite recursion as follows.

- $x_0=x$,
- $x_{\alpha+1}=g_{\alpha}\cdot x_{\alpha}$ for all ordinals $\alpha<\lambda$, and
- $x_{\theta}=\begin{cases} \ast & \text{ if } \lim_{\alpha \rightarrow \theta} x_{\alpha} \text{ does not exist}\\ \lim_{\alpha \rightarrow \theta} x_{\alpha} & \text{ otherwise }\end{cases}$

for every limit ordinal $\theta \leq \lambda$. Let us say that a sequence $(g_{\alpha})_{\alpha < \lambda}$ is **valid** if $x_{\lambda} \neq \ast$ for all $x \in X$. In other words, valid sequences are those along which every point can be transfinitely iterated.

For each valid sequence $\mathbf{g}=(g_{\alpha})_{\alpha < \lambda}$, consider the map $K_{\mathbf{g}}: X \rightarrow X$ given by $K_{\mathbf{g}}(x)=x_{\lambda}$.

Here are my questions.

Under what (not very restrictive) conditions can we guarantee that the set of $K=\{K_{\mathbf{g}}: \mathbf{g} \text{ is valid of length} <\omega_1\}$ forms a subgroup of $\Sym(X)$?

Can we see this set as some kind of "closure" of the image of $G$ in $\Sym(X)$?

Here are some basic facts that I was able to show

- The maps $K_{\mathbf{g}}$ are not necessarily bijections. For example, consider the shift action of $\mathbb{Z}$ on the set $X$ of bi-infinite 0-1 sequences that are constant after some index. Then all sequences are mapped to constant sequences under the maps corresponding to the valid sequence $(1,1,1\dots)$ of length $\omega$.
- If $G$ acts on a compact metric space $X$ by isometries, then each $K_{\mathbf{g}}$ is an isometry. Preservation of distances is trivial and surjectivity can be shown with some effort using transfinite induction and the compactness of the space. However, I cannot show that $K_{\mathbf{g}}^{-1}$ is of the form $K_{\mathbf{h}}$ and so, I cannot guarantee that these maps form a subgroup.

For an example of a non-trivial case where the maps $K_{\mathbf{g}}$ give us more bijections of $X$ than those induced by $G$, consider the action of $\mathbb{Q}$ on $S^1$ by rotations. In this case, the maps $K_{\mathbf{g}}$ induced by valid sequences of length $\omega$ gives all the **real** rotations because for any real $r$, we can form a sequence $(q_0,q_1,\dots)$ of rationals with $\sum_{i=0}^{\infty} q_i=r$. On the other hand, no other valid sequence gives us a bijection other than these real rotations.

Iterating such a system along ordinals seems very natural to me, however, I was not able to find much on this. As I said, if anything along these lines were investigated before, I would appreciate being directed to the correct references.

Ellis semigroupof the action is the closure of (the image of) $G$ in $X^X$. For instance, one can characterize when it's indeed a group of homeomorphisms. In opposite cases such as convergence actions, it consists of $G$ union functions that are constant outside a singleton. $\endgroup$