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 C α $\mathcal {C}^\alpha $ script upper C Superscript alpha -stably exhibit fiberwise bounded orbits ( α > 0 $\alpha>0$ alpha greater than 0 ). The proof is based on a new mechanism which yields stable elliptic-type behavior in GL ( d , R ) $\mathrm {GL}(d,\mathbb {R})$ upper G upper L left parenthesis d comma double struck upper R right parenthesis or SL ( d , R ) $\mathrm {SL}(d,\mathbb {R})$ upper S upper L left parenthesis d comma double struck upper R right parenthesis cocycles. Moreover, we show that this phenomenon is C 0 $\mathcal {C}^0$ script upper C Superscript 0 -dense among SL ( d , R ) $\mathrm {SL}(d,\mathbb {R})$ upper S upper L left parenthesis d comma double struck upper R right parenthesis cocycles over a shift of finite type without dominated splitting.