This paper considers skew-product systems and skew-product flows. The author defines a skew-product system as follows: consider real analytic vector fields on \({\mathcal M}= \mathbb{T}^d\times G\) (where \(\mathbb{T}= \mathbb{R}/2\pi\mathbb{Z}\) and \(G\subset\text{GL}(n,\mathbb{R})\) is a Lie group with Lie algebra \(g\)) defined by \(X(x,y)= (\omega, f(x)y)\) with \(\omega\in\mathbb{R}^d\) and \(f:\mathbb{T}^d\to g\). These generate differential equations \[ {dx\over dt}= \omega\quad\text{and}\quad {dy\over dt}= f(x)y,\tag{\(*\)} \] where \((x,y)\in{\mathcal M}\). The author assumes that \(\omega\cdot\nu=\) all \(\nu\in\mathbb{Z}^d- \{0\}\). The skew-product flow is then \[ (x,y)\mapsto (x+\omega t,\Phi^t(x)y), \] where \(\Phi^+(x)\) is the matrix solution of the system \((*)\) for which \(\Phi^0(x)\) is the identity on \(G\).
The problem of reducibility is the question of whether there is a coordinate change \(z= \psi^{-1}(x)y\) such that system \((*)\) transforms to \[ {dx\over dt}= \omega\quad\text{and}\quad {dz\over dt}= uz. \] In this paper, the author assumes that the vector \(\omega\) is Brjuno so that \[ \sum^\infty_{n=1} 2^{-n}\ln(1/\Omega_n), \] where \(\Omega_n\) is the minimum of \(|\omega\cdot\nu|\) for \(\nu\in\mathbb{Z}^d\) and \(|\nu|< 0\leq 2^n\).
The author develops a renormalization method that permits local reducibility when the group \(G\) above is \(\text{SL}(2,\mathbb{R})\) and then applies the result to prove a reducibility theorem for flows, where \(\omega\) is a Brjuno vector. This approach – an alternative to the KAM theory for small divisors – rescales the torus to turn small divisors into large divisors that can be eliminated by change of coordinates.