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.

On the spectrum of the spectrum of the periodic anderson-bernoulli model

“…First, we will need to define what it means that a cocycle or a set of matrices is uniformly hyperbolic. (The notion of uniform hyperbolicity in this paper largely references [1], as well as [12] and [3]). There are two contexts over which uniform hyperbolicity can be defined.…”

“…It would seem possible that for a large enough period, the spectrum may consist of an infinite number of intervals. Further research in this field can be seen in [3].…”