COCYCLES WITH QUASI-CONFORMALITY I: STABILITY AND ABUNDANCE

Abstract 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 $\mathrm {GL}(d,\mathbb {R})$ or $\mathrm {SL}(d,\mathbb {R})$ cocycles. Moreover, we show that this phenomenon is $\mathcal {C}^0$ -dense among $\mathrm {SL}(d,\mathbb {R})$ cocycles over a shift of finite type without dominated splitting.