Furstenberg’s Correspondence Principle
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Theorem 1 (cf. Kra et al. (2022), Theorem 2.10) Let \(\AmenableGroup\) be an amenable group, \(A\subset \AmenableGroup\), and \(\Folner\) be a Følner sequence in \(\AmenableGroup\) such that the limit \[\delta=\lim_{N\rightarrow\infty}\frac{\CountingMeasure{A\cap\Folner[N]}}{\CountingMeasure{\Folner[N]}}\] exists.
Then there exists an ergodic system \((\ShiftSpace,\Measure,\AmenableGroup)\) that is acted on by \(\AmenableGroup\), a clopen set \(E\subset\ShiftSpace\), a Følner sequence \(\FurstenbergFolner\) in \(\AmenableGroup\), and a point \(a\in\ShiftSpace\) that is generic with \(\Measure\) with respect to \(\FurstenbergFolner\) such that \(\Measure(E)\geq\delta\) and \[A={\AmenableGroupElement\in \AmenableGroup:\GroupAction{\AmenableGroupElement}{a}\in E}. \]