Lifting ergodicity in \((G\sigma)\)-extensions (Q2475548)

Let \((X,T,m)\) be a dynamical system, where \(X\) is a compact metric space, \(m\) is a probability measure and \(T\) is a measure preserving transformation. Let \(G\) be a compact metric topological group with metric \(d\) and \(\sigma\) be a continuous group automorphism of \(G\). Given a continuous map \(\varphi: X\to G\), define a \(\sigma\)-skew product transformation \(T_\varphi: G\times X\to G\times X\) by \[ T_\varphi(g,x)=(\sigma[\varphi(x)g],Ty),\quad(g,x)\in G\times X. \] Let \(\nu\) denote the normalized Haar measure on \(G\). Denote \[ C_{erg}(X,G)=\{\varphi\in C(X,G): (G\times X,T_\varphi,\nu\times m)\hbox{ is ergodic}\}. \] The pair \((G,\sigma)\) has Property \textbf{A} if given a neighborhood \(W\) of the identity \(\epsilon_G\), there exists a positive integer \(\kappa\) such that, if \(W_0,\ldots,W_{\kappa-1}\) are any \(\kappa\) right translates of \(W\), then \[ \prod_{i=0}^{\kappa-1}\sigma^{\kappa-i}(W_i)=G. \] The pair \((G,\sigma)\) has Property \textbf{B} if given a compact metric space \(X\) and a \(\varphi\in C(X,G)\) and \(\varepsilon>0\), there exists a \(\delta>0\) such that if \(F\) is a finite subset of \(X\) and \(\psi: F\to G\) is a function that satisfies \(d(\varphi(x),\psi(x))<\delta\) for all \(x\in F\), then \(\psi\) can be extended to a continuous map on \(X\) such that \(d((\varphi(x),\psi(x))<\varepsilon\) for all \(x\in X\). A pair \((G,\sigma)\) is called \textit{admissible} if it satisfies both Property \textbf{A} and Property \textbf{B}. Proposition 1.2. If \(G\) is either (i) a compact, metric connected abelian group or (ii) a compact, connected Lie group or (iii) a compact connected metric group with finite center, then \((G,\sigma)\) is admissible for any continuous automorphism \(\sigma\). The main theorem of this article is: Theorem 1.3. Suppose that (i) \((X,T,m)\) is an ergodic and aperiodic, and (ii) \((G,\sigma)\) is admissible. Then the set \(C_{erg}(X,G)\) is a residual subset of \(C(X,G)\).