Let \((\Omega, \{T_ t\}_{t \in \mathbb{R}})\) be a flow on a compact metric space \(\Omega\). Given a cocycle \(\Phi : \Omega \times \mathbb{R} \to \text{GL}(n, \mathbb{R})\), consider the skew product extension \((\mathbb{R}^ n \times \Omega, \{\widehat{T}_ t\}_{t \in \mathbb{R}})\), where \(\widehat{T}_ t(v,\omega) = (\Phi (\omega, t)v, T_ t \omega)\). Let \(B = \{(v,\omega) \in \mathbb{R}^ n \times \Omega \mid\) the set \(\{\Phi(\omega, t)v \mid t \in \mathbb{R}\}\) is bounded}. In general \(B\) is not a vector subbundle of the vector bundle \(\mathbb{R}^ n \times \Omega\), and it is well known that it is so if and only if \(B\) satisfies the “Faward property”.
In this paper the author considers a subset \(D\) of \(B\) defined by \(D = \{(v,\omega) \in B \mid\) the orbit closure of \((v,\omega)\) under the skew product flow is a distal extension of the base flow}.
The main theorem of the paper says that the set \(D\) is a subbundle if the base flow is minimal.