Recurrence and Ergodic Theorems
Notation
| Term | Description |
|---|---|
| \(\N\) | The Natural Numbers: 1, 2, 3, … (Source) |
Backlinks
Theorem 1 (Bergelson, 1985, Theorem 1.1) Let \((X,\SigmaAlgebra{B},\Measure)\) be a probability space and suppose that \(B_n\in\SigmaAlgebra{B}\) such that \(\Measure(B_n)=b>0\) for all \(n\in\mathbb{N}\).
Then there exists a positively dense index set \(I\subset\mathbb{N}\) such that, for any finite subset \(F\subseteq I\), we have \[\Measure\left(\bigcap_{i\in F}B_i \right)>0. \]
Theorem 2 (Poincaré Recurrence for Upper Density) Let \(\Monoid\) be a countably-infinite amenable group with \(\Folner\) as a Følner sequence. For all \(B\in\mathcal{P}(\Group)\) such that \(\Density{B}>0\), there exists \(\GroupElement\in\Group\) such that \(\UpperDensity{B\cap \GroupOperation{\GroupElement}{B}}>0\).
This can be proved by using Furstenberg’s Correspondence Principle with \(B\) as our subset and, as \(\Measure\) is measure-preserving in our Furstenberg system and by using the invariance of density, we can show that \[\Density{B}=\Measure(E)=\Measure(\GroupAction{\GroupElement}{E})=\Density(\GroupOperation{\GroupElement}{B})\] for all \(\GroupElement\in\Group\). By defining an infinite series \(E_n=\GroupAction{g_n}{E}\) and knowing that \(\Measure(E_n)>0\) for all \(n\in\N\), we can use Theorem 1 to arrive at the result that \(\Measure(\GroupAction{h}{B}\cap \GroupAction{h'}{B})>0\) for some \(h,h'\in\Group\). By using the fact that \(\Measure\) is measure-preserving again and using the properties of \(\Measure\) as a result of its construction in Furstenberg’s Correspondence Principle, we conclude that \(\UpperDensity(B\cap\GroupOperation{g}{B})>0\) for some \(\GroupElement\in\Group\)
Theorem 3 (cf. Lindenstrauss, 2001, Theorem 1.2) Let \(\AmenableGroup\) be a discrete amenable group acting on a measure space \((X,\SigmaAlgebra{B},\Measure)\) by measure preserving transformation and let \(\Folner=(\Folner[N])_{N\in\mathbb{N}}\) be a tempered Følner sequence.
Then, for any \(f\in\text{L}^1(\Measure)\), there is a \(\AmenableGroup\)-invariant \(\bar{f}\in\text{L}^1(\Measure)\) such that \[\lim_{N\rightarrow\infty}\frac{1}{\CountingMeasure{\Folner[N]}}\sum_{\AmenableGroupElement\in\Folner[N]}f(\GroupAction{\AmenableGroupElement}{x})=\bar{f}(x) \] for \(\Measure\)-almost every \(x\in X\). In particular, if the \(\AmenableGroup\) action is ergodic, then \[\lim_{N\rightarrow\infty}\frac{1}{\CountingMeasure{\Folner[N]}}\sum_{\AmenableGroupElement\in\Folner[N]}f(\GroupAction{\AmenableGroupElement}{x})=\int f(x)\ d\Measure(x) \] for \(\Measure\) almost every \(x\).
Corollary 1 (cf. Host, 2019, Corollary 8) Let \((X,\AmenableGroup)\) be a topological dynamical system where \(\AmenableGroup\) is an amenable group, \(\Measure\) an ergodic measure on \(X\) and \(\Folner\) a tempered Følner sequence. Then \(\Measure\)-almost every \(x\in X\) is generic for \(\Measure\) along \(\Folner\).
Theorem 4 (Jamneshan and Kreidler, 2025, Abstract Mean Ergodic Theorem) For every contraction semi-group \(\mathscr{S}\subseteq\mathscr{L}(H)\) on a Hilbert space \(H\), we have an orthogonal decomposition \[H=\text{fix}(\mathscr{S})\oplus\overline{\text{lin}}\bigcup_{U\in\mathscr{S}}(\text{Id}_H-U)(H).\] Moreover, the orthogonal projection \(P\) onto \(\text{fix}(\mathscr{S})\) has the following properties:
- \(PU=UP=P\) for every \(U\in\mathscr{S}\).
- For every \(f\in H\), the vector \(Pf\in H\) is the unique element of \(\overline{\text{co}}\{Uf\mid U\in\mathscr{S}\}\) of minimal norm.