Let $(G,X,\pi)$ be a transformation group, i.e. $G$ a topological group, $X$ a compact Hausdorff space, and $\pi:G\times X\to X$ a continuous group action. Let $E:= E(G,X)$ be the closure of $\{\pi(g,\cdot) : g\in G\}$ in $X^X$ (with the product topology).
Now, Ellis (in Lectures on Topological Dynamics) and Auslander (in Minimal Flows And Their Extensions) both claim that $(G,E,\tilde \pi)$ with $$\tilde \pi : G\times E \to E,\qquad (g,f)\mapsto \pi(g,\cdot)\circ f$$ is again a transformation group. However, they don't go into detail as to why $\tilde \pi$ is continuous.
Ellis claims to prove continuity by taking two nets $(g_\alpha)_\alpha$ in $G$ and $(f_\beta)_\beta$ in $E$ and then argues $$\lim_\beta\lim_\alpha \tilde \pi(g_\alpha, f_\beta) = \lim_\beta \tilde\pi(g,f_\beta) = \tilde \pi(g,f).$$ I understand the identity but I fail to see why this proves continuity of $\tilde \pi$. In my opinion one has to take one net $(g_\alpha,f_\alpha)_\alpha$ in $G\times E$ that converges to $(g,f)$ and show that $$\lim_\alpha \tilde\pi (g_\alpha, f_\alpha) = \tilde\pi (g,f).$$
So, my two questions are:
- Am I missing something here? Does the argument hold?
- If it does not hold, how does one prove the continuity of $\tilde \pi$.
It is even continuous as a map $G\times X^X\to X^X$.
Consider $g_i\to g$, $f_i\to f$, with $f_i,f\in X^X$. So for every $x\in X$, $f_i(x)\to f(x)$.
You need to show that $\pi(g_i,\cdot)\circ f_i$ pointwise converges to $\pi(g,\cdot)\circ f$. That is, for every $x$ you need to show that $\pi(g_i,\cdot)\circ f_i(x)=\pi(g_i,f_i(x))$ pointwise converges to $\pi(g,\cdot)\circ f(x)=\pi(g,f(x))$. But this is just what continuity of $\pi$ at $(g,f(x))$ says.