Cocycles with Quasi-Conformality I: Stability and abundance

TL;DR

This paper introduces a new mechanism for constructing linear cocycles over chaotic systems that exhibit stable fiberwise bounded orbits, demonstrating their abundance and stability in certain dynamical contexts.

Contribution

It establishes the existence and density of $ ext{C}^eta$-stable fiberwise bounded cocycles with elliptic behavior in $ ext{GL}(d, ext{R})$ and $ ext{SL}(d, ext{R})$, over shifts of finite type.

Findings

Existence of $ ext{C}^eta$-stable fiberwise bounded cocycles.

Density of such cocycles among $ ext{SL}(d, ext{R})$ cocycles.

Introduction of a new mechanism for stable elliptic behavior.

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.