# Group actions and "transfinite dynamics"

$$\DeclareMathOperator\Sym{Sym}$$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.

• Although distinct, let me point out a slightly related well-studied concept: let $G$ act on a compact space $X$ (usually by homeomorphisms). Let $X^X$ be endowed with the product compact topology. The Ellis semigroup of 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.
– YCor
Feb 10, 2020 at 14:12
• You're idea reminds me of this paper by Küster where she relates the fixed space of the Koopman operator on a topological dynamical system to transfinite properties of the orbits of the underlying dynamic. Feb 10, 2020 at 20:07

For every sequence $$\mathbf{g}=(g_{\alpha})_{\alpha < \lambda}$$ and every $$i \in \mathbb{N}$$, set $$\mathbf{g}^i$$ to be the sequence of length $$\lambda \cdot i$$ given by $$g^i_{\lambda j + \alpha}=g_{\alpha}$$ for all $$\alpha < \lambda$$ and $$j < i$$. In other words, $$\mathbf{g}^i$$ is the concatenation of $$i$$-many $$\mathbf{g}$$'s back to back. Observe that $$\mathbf{g}^i$$ is valid whenever $$\mathbf{g}$$ is valid. Moreover, one can show by induction that for every valid sequence $$\mathbf{g}$$ and every $$i \in \mathbb{N}$$, we have $$K_{\mathbf{g}}^i=K_{\mathbf{g}^i}$$.
Lemma: Let $$\mathbf{h}$$ be a valid sequence such that $$K_{\mathbf{h}}: X \rightarrow X$$ is a bijection. If there exists $$k \in \mathbb{N}$$ such that $$|\text{Orb}(x)| \leq k$$ for all $$x \in X$$, then the map $$K_{\mathbf{h}^{k!-1}}$$ is the inverse of $$K_{\mathbf{h}}$$.
Proof: Assume that $$k \in \mathbb{N}$$ is a uniform bound for orbit sizes. It follows from the finiteness of orbits that the trajectory of a point under $$\mathbf{h}$$ must be eventually constant and hence, we must have $$K_{\mathbf{h}}(x) \in \text{Orb}(x)$$ for every $$x \in X$$. Consequently, $$K_{\mathbf{h}} \upharpoonright \text{Orb}(x) \in \text{Sym}(\text{Orb}(x))$$ for every $$x \in X$$. By Lagrange's theorem, $$\left(K_{\mathbf{h}} \upharpoonright \text{Orb}(x) \right)^{k!} = \mathbf{id}_{\text{Orb}(x)}$$ for every $$x \in X$$ and hence $$K_{\mathbf{h}}^{k!} = \mathbf{id}_X$$. By the previous lemma, $$K^{k!-1}_{\mathbf{h}}=K_{\mathbf{h}^{k!-1}}$$ is the inverse of $$K_{\mathbf{h}}$$.