Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups (Q2831666)

Cocycles considered in this memoir are discrete dynamical systems, whose phase space is a fibered space \(X \times E \to X\). A fibered dynamics is defined by the iteration of a mapping of the type \((T, f): X \times E \to X \times E\), where \(T\) is a mapping of \(X\) into itself, and \(f : X \times E \to E\) is some mapping. If \(E\) is a group \(G\) or a space on which a group acts, this kind of fibered dynamics is called a cocycle.NEWLINENEWLINEA particular case occurs when \(X = \mathbb T^d = \mathbb R^d/\mathbb Z^d\) (a \(d\)-dimensional torus), and \(T = R_\alpha : x \to x + \alpha\), \(\alpha =(\alpha_1, \dots ,\alpha_d)\), is a minimal translation (it is equivalent to \(\alpha_i\) being linearly independent over \(\mathbb Z\)), and therefore ergodic with respect to the Haar measure on the torus. Here a cocycle acts on the fibered space as follows. For a smooth mapping \(A:\mathbb T^d \to G\) the pair \((\alpha, A)\) acts as a diffeomorphism by \((\alpha, A)(x, S) = (x + \alpha, A(x)S)\) for any \((x, S) \in \mathbb T^d \times G\). These cocycles are called quasiperiodic and \(\alpha =(\alpha_1, \dots ,\alpha_d)\) is called the frequency of the cocycle, \(d\) is the number of frequencies. When \(d=1\) and \(\alpha\) is rational, cocycles \(R_\alpha\) are perfectly understood due to the Floquet representation of solutions. For cocycles, the Floquet representation of a solution corresponds to conjugation of a cocycle to a constant one. The main goal is to give some analogue of Floquet theory in a general situation, in particular to examine the density properties of Floquet-type solutions, and the possibility of approximation of any given vector field with a field admitting Floquet-type solutions.NEWLINENEWLINEThe author studies quasiperiodic cocycles in semisimple compact Lie groups \(G\). For his study, he focuses on one-frequency cocycles (\(d=1\)) because not all of his proofs work when \(d \geq 2\). In this case minimality of translation is equivalent to \(\alpha\) being an irrational number.NEWLINENEWLINEThe cocycle \((R_\alpha, f)\) is called reducible if \(A\) is homotopic to a constant mapping \(\mathbb T^n \to G\). It is proved that \(C^\infty\)-reducible cocycles are dense in the \(C^\infty\)-topology, for a full-measure set of frequencies \(\alpha\). Moreover, it is shown that every cocycle (or an appropriate iterate of it, if some homotopy appears as an obstruction) is almost torus-reducible (i.e., it can be conjugated arbitrarily close to cocycles taking values in an abelian subgroup of \(G\)).NEWLINENEWLINEThe author defines and investigates two invariants of the dynamics, which he calls energy and degree and which give a preliminary distinction between (almost-)reducible and non-reducible cocycles. Then he proves a density theorem and shows that an algorithm of renormalization converges to perturbations of simple models, indexed by the degree. Finally, he analyzes these perturbations using methods inspired by KAM theory.NEWLINENEWLINEA concrete example with \(d=1\) and \(G= {\mathrm{SU}}(3)\) is considered.