Kronecker Factor
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Definition 1 (Jamneshan and Kreidler, 2025, Definition 7.1.14 & Subsection 13.3) Let \((X,T)\) be a measure-preserving system. We then write \(J_\text{kro}:(X_\text{kro},T_\text{kro})\rightarrow(X,T)\) for the extension \(J_E:(X_E,T_E)\rightarrow(X,T)\) defined by the invariant Markov sublattice \(E=\text{L}^2(X)_\text{ds}\) and call \((X_\text{kro},T_\text{kro})\) the Kronecker subsystem of \((X,T)\).
We refer to \((X_\text{kro},\Topology_\text{kro})\) as the Kronecker factor of the system \((X,\Topology)\).
Proposition 1 (cf. Host, 2019, Proposition 5) Let \((X,\AmenableGroup)\) be a topological dynamical system where \(\AmenableGroup\) is an amenable group, \(x_0\in X\), and \(\Measure\) be an ergodic invariant probability measure supported on the closed orbit of \(x_0\) under the action of \(\AmenableGroup\).
Let \(\KroneckerFactor\) be the Kronecker factor of \((X,\Measure,\AmenableGroup)\), with factor map \(\ProjectionMap:X\rightarrow\KroneckerSpace\).
Let \(X\times\KroneckerSpace\) be endowed with the group action of \(\AmenableGroup\times\KroneckerAction\). Let \(\tilde\Measure\) be the measure on \(X\times\KroneckerSpace\) and image of \(\Measure\) under the map \(X\rightarrow X\times\KroneckerSpace\) where \(x\mapsto(x,\ProjectionMap(x))\).
Then there exists a Følner sequence \(\tilde{\Folner}\) and a point \(\KroneckerSpaceElement_0\in\KroneckerSpace\) such that \((x_0,\KroneckerSpaceElement_0)\) is generic for \(\tilde{\Measure}\) along \(\tilde{\Folner}\).
The sources given by Host (2019) for this result were Kra and Host (2007), Proposition 6.1 and Host and Kra (2018), Proposition 24.3.