Parameter Spaces of Locally Constant Cocycles

Abstract This article concerns the locus of locally constant $\textrm{SL}(2,\mathbb{R})$-valued cocycles that have a dominated splitting, called the hyperbolic locus. By developing the theory of Möbius semigroups we show that cocycles on the boundary of the hyperbolic locus, apart from a few exceptions, exhibit some form of hyperbolic behaviour. This behaviour is used to answer a question posed by Avila, Bochi and Yoccoz. Our approach introduces a new locus of cocycles, closely related to the hyperbolic locus, and motivates a line of investigation on the subject.