Factor Maps
Backlinks
Definition 1 (Kra et al. (2022), Definition 2.1) For a system \((X,\Measure,T)\) we say that the system \((Y,\ u,S)\) is a measurable factor of \((X,\Measure,T)\) if there is a measurable map \(\ProjectionMap:X\rightarrow Y\), the measurable factor map, such that \(\ProjectionMap(\Measure)=\ u\) and \((S\circ\ProjectionMap)(x)=(\ProjectionMap\circ T)(x)\) for \(\Measure\)-almost every \(x\in X\).
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 \(\tilde{\Folner}\).