Notes about math, research, and more.
:tags: topology, flows

Definition

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

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

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

Synonyms:

  1. transformation group ↔ flow
  2. topological space ↔ phase space
  3. topological group ↔ phase group or acting group

Typical assumptions

  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.

Maps

\(\pi^t\)

Each \(t \in T\) defines a continuous map \(\pi^t\) of \(X \to X\) by \[ \pi^t(x) = \pi(x,t). \] If \(t,s \in T\), it is immediate that \(\pi^s \pi^t = \pi^{st}\); in particular, \(\pi^{t^{-1}}\pi^t = \pi^e\), the identity map of \(X\), so each \(\pi^t\) is a homeomorphism of \(X\) onto itself, with \(\pi^{t^{-1}} = (\pi^t)^{-1}\).

Axioms

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

Notation

  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 (\(\pi^t\)).

Notes

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