Category theorems for some ergodic multiplier properties (Q1082469)

Let G be the group of nonsingular transformations T of \([0,1]\) (with Lebesgue measure m), and let \(G_ m\) be the subset of those T preserving m. For \(P\subset G\) let \( E(P)=\{S\in G_ m:S\times T\) is ergodic for all \(T\in P\}.\) The author studies the category of E(P) in \(G_ m\) for certain P. For a function \(\rho(t)>0\) with \(\rho(t)\downarrow 0\) and \(\rho(t)/t\uparrow \infty ast\downarrow 0\), let \(G^ e(\rho)\) denote the set of ergodic T's in G for which the eigenvalue group has \(\rho\)- Hausdorff measure 0. It is shown that \(E(G^ e(\rho))\) is meagre in \(G_ m\) (with the weak topology). For a decreasing sequence \(c_ n\downarrow 0\) with \(\sum c_ n=\infty\) let \(g^ e((c_ n))\) be the set of ``\(c_ n\)-recurrent'' \(T\in G\). Also \(E(G^ e(c_ n))\) is meagre in \(G_ m\).