A transformation group is a triple $(X,T,\pi )$ where

• $X$ is a topological space,
• $T$ is a topological-group, and
• $\pi$ is a continuous map of $X\times T\to X$.

$(X,T,\pi )$ satisfy the following:

1. $\pi (x,e)=x(x\in X,e\text{ the identity of }T)$ and
2. $\pi (\pi (x,s),t)=\pi (x,st)(x\in X,s,t\in T)$.

1. The phase space $X$ is Hausdorff.
2. Often assume the topology on $T$ is discrete (because the topology is mostly irrelevant).
3. Typically assume that $X$ is compact.

1. $(x,t)\mapsto xt$ is continuous
2. $xe=x$
3. $(xs)t=x(st)$

Let $(X,T)$ and $(Y,T)$ be flows with the same acting group. A homomorphism from $X$ to $Y$ is a continuous map $\pi :X\to Y$ such that $\pi (xt)=\pi (x)t$ for all $x\in X,t\in T$.

If there is a homomorphism $\pi$ from $X$ onto $Y$, we say that $T$ is a factor of $X$ and $X$ is an extension of $Y$.

1. flows will be used over transformation group
2. $\pi (x,y)$ will be written as $xt$
3. Regard $t\in T$ as “acting on $X$”.
4. Usually write $(X,T)$ or even just $X$ if the group $T$ is understood.
5. An element $t\in T$ will be identified with the homeomorphism of $X$ it defines (${\pi }^t$).

The topology of the group is really not that important - it is the action of the group of homeomorphisms that is interesting.

1. equicontinuous flow
2. flow
3. book-notes-minimal-flows-and-their-extensions