Cocycles with Quasi-Conformality I: Stability and abundance

This is the first part of a series of papers devoted to the study of linear cocycles over chaotic systems. In the present paper, we establish the existence of such cocycles that ${\mathcal{C}}^{\alpha }$-stably exhibit fiberwise bounded orbits ($\alpha >0$). The proof is based on a new mechanism which yields stable elliptic-type behavior in $\mathrm{G}\mathrm{L}(d,\mathbb{R})$ or $\mathrm{S}\mathrm{L}(d,\mathbb{R})$ cocycles. Moreover, we show that this phenomenon is ${\mathcal{C}}^0$-dense among $\mathrm{S}\mathrm{L}(d,\mathbb{R})$ cocycles over a shift of finite type without dominated splitting.