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Definition 1 (Bekka and Mayer (2000) Section 2) An action of a group, \(\Group\), on a measurable space \((X,\SigmaAlgebra{B})\) is a measurable mapping \[\Group\times X\rightarrow X,\ (\GroupElement,x)\mapsto \GroupAction{\GroupElement}{x} \] with the following properties:
- Associativity: For all \(\GroupElement,\GroupElement'\in \Group,x\in X\), then \(\GroupAction{\GroupElement}{(\GroupAction{\GroupElement'}{x})}=\GroupAction{(\GroupOperation{\GroupElement}{\GroupElement'})}{x}\)
- Identity: There exists an identity element \(\GroupIdentity\in\Group\) such that \(\GroupAction{\GroupIdentity}{x}=x\) for all \(x\in X\).
- Quasi-Invariance: For any \(B\in\SigmaAlgebra{B}\) and for all \(\GroupElement\in\Group\), we have \(\Measure(\GroupAction{\GroupElement}{B})=0\) if and only if \(\Measure(B)=0\).
The action of \(\Group\) is also ergodic if it satisfies the additional property:
- If \(B\in\SigmaAlgebra{B}\) and \(\Measure(B)=\Measure(\GroupAction{\GroupElement}{B})\) for any \(\GroupElement\in\Group\), then \(\Measure(B)=0\) or \(\Measure(X\setminus B)=0\).
Remark 1. We don’t require invertability in order to use actions and could instead use a monoid, \(\Monoid\), defining the pre-image of \(\Monoid\) on \((X,\SigmaAlgebra{B})\) as a measurable mapping \[\Monoid\times X\rightarrow \SigmaAlgebra{B},\ (\MonoidElement,x)\mapsto \GroupActionPreImage{\MonoidElement}{x} \] such that \[\GroupActionPreImage{\MonoidElement}{x}=\{x'\in X: \GroupAction{\MonoidElement}{x'}=x \}. \]
Definition 2 A topological dynamical system under the action of \(\Group\), denoted \((X,\Group)\), is a compact metric space \(X\) that has continuous surjective maps, \((\GroupElement,x)\mapsto \GroupAction{\GroupElement}{x}\), for all \(\GroupElement\in\Group\).
Definition 3 Let \(x\in X\), \(\Folner=(\Folner[N])_{N\in\mathbb{N}}\) be a Følner sequence in \(\Gamma\) and \(\Measure\) a probability measure on \(X\). Where \(\delta_x\) is the Dirac mass at \(x\), if \[\frac{1}{|\Folner[N]|}\sum_{\GroupElement\in\Folner[N]}\delta_{\GroupAction{\GroupElement}{x}}\underset{\text{weakly*}}{\longrightarrow} \Measure \text{ as }N\rightarrow\infty, \] then we say \(x\) is generic for \(\Measure\) with respect to \(\Folner\) and we denote this with \(x\in\text{gen}(\Measure,\Folner)\).1
We are interested in how the action of a group \(\Group\) transforms functions on \((X,\SigmaAlgebra{B},\Measure)\) so we must identify the associated definitions within functional analysis.
We define the map \(\KoopmanOperator:\text{L}^2(X)\rightarrow\text{L}^2(X)\) where \(f\mapsto f\circ \KoopmanOperator\) as the Koopman operator induced by \(\GroupElement\in\Group\). We find that the Koopman operator induced by \(\Group\) is a unitary operator and the group homomorphism \(U:\Group\rightarrow\mathscr{U}(\text{L}^2(X))\) is the unitary representation of \(\Group\) on \(\text{L}^2(X)\), where \(\mathscr{U}(\text{L}^2(X))\) is the set of all unitary operators on \(\text{L}^2(X)\).