Minimal acceleration and the topology of premonotonic cocycles

In his global theory of one-frequency Schrödinger operators~\cite{Av15}, Avila asked whether the acceleration of a real-analytic $\SL(2,\R)$ cocycle over an irrational circle rotation is bounded below by the absolute value of its topological degree and whether equality in nonzero degree characterizes premonotonicity. We answer this question affirmatively for every irrational frequency, without arithmetic restrictions. Here premonotonicity means that some iterate becomes strictly monotonic in the base variable after a periodic change of coordinates~\cite{AK15}. In the positive Lyapunov exponent regime, this provides a nonzero-degree counterpart to Avila's characterization of uniform hyperbolicity by vanishing acceleration. In nonzero degree, we also characterize minimal acceleration through signed connections and invariant positive matrix-valued measures. In the positive-degree, positive Lyapunov exponent case, the proof constructs an analytic positive connection using invariant positive matrix-valued measures and a transfer--scattering representation. A complex-analytic construction then produces a periodic gauge. As a further consequence, for each fixed irrational frequency and nonzero degree, the premonotonic locus is path connected: every cocycle in this locus can be joined to the linear rotation of the same degree by a path preserving premonotonicity.