Cocycles with Quasi-Conformality II: Ergodic measures with positive entropy

Abstract

As the second part of a series on linear cocycles over chaotic systems, this paper establishes a "multiple covering principle" that robustly yields positive-entropy ergodic measures supported on fiberwise uniformly bounded orbits. Using this mechanism, we prove that any continuous $\mathrm{SL}(d,\mathbb{R})$ cocycle over a positive-entropy subshift of finite type either admits a dominated splitting or can be ${\mathcal{C}}^0$-approximated by one that \emph{stably} supports such measures. The stability holds in Hölder topology which is optimal. Additionally, for non-isometric cocycles, we show that the topological entropy of these bounded orbits is strictly less than that of the base subshift.