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Chapter 1: Flows and Minimal Sets

Defines flow.

Conventions

  1. Transformation group will be refered to as flows.
  2. Each \(t \in T\) defines a continuous map \(\pi^t:X \to X\) by \(\pi^t(x) = \pi(x,t)\). Will usually suppress the map \(\pi\) and just write \(xt\) instead of \(\pi^t(x)\).
  3. Identify \(t \in T\) with the homeomorphism of \(X\) it defines. Thus, \(T\) may be regarded as a subgroup of the total homeomorphism group of \(X\). It is possible distinct \(s,t \in T\) define the same homeomorphism.
  4. \(X\) is Hausdorff
  5. Action on \(X\) is effective (see blow for definition and reason).
The action of \(T\) is called effective if \(t\neq e\) implies \(xt\neq x\) for some \(x \in X\). (The homeomorphisms \(T\) induces are unique).

If the action is not effective then the group \(F = \{ t \in T \mid xt=x ~\forall x \in X \}\) is a normal subgroup of \(T\) (since \(X\) assumed to be hausdorff). The quotient \(T/F\) acts on \(X\) by \(x(Ft) = xt\) which is effective.

Pg 3: Topology on \(T\) is irrelevant Even definitions which seem to depend on the topology of \(T\) will be shown to be independent of it.

\(X\) a flow with transformation group \(T=\mathbb{Z}\). If \(\phi\) is the homeomorphism \(\phi(x) = x1\) then \(\phi^n(x) = xn\) and so \((X,\phi)\) is a cascade: the flow \(X\) along with \(\phi\) and its powers.
If \(T = \mathbb{R}\) then the action defines a one-parameter group of homeomorphisms \(\{ \phi_{t} \}\) of the space \(X\) and we write \(\phi_{t}(x)\) instead of \(xt\). This flow is a dynamical system

If \(\phi:X \to X\) is a continuous map of a compact space, it can be converted to a homeomorphism by an “inverse limit” construction. Let \(X^\mathbb{Z}\) denote the sapce of 2-sided sequences of elements of \(X\), with the product topology, and let \(\tilde{X}\) denote the subset of \(X^\mathbb{Z}\) consisting of those sequences \(x=(x_{1})\) for which \(\phi(x_{i+1}) = x_{i} ~\forall i \in \mathbb{Z}\). It follows easily from the compactness of \(X\) that \(\tilde{X}\) is a non-empty compact space.