Zahra Reshadat | Searcharxiv

arXiv2026

Cocycles with Quasi-Conformality II: Ergodic measures with positive entropy

As the second part of a series on linear cocycles over chaotic systems, this paper establishes a "multiple covering principle" that robustly yields positive-entropy ergodic measures supported on fiberwise uniformly bounded orbits. Using this mechanism, we prove that any continuous $\mathrm{SL}(d,\mathbb{R})$ cocycle over a positive-entropy subshift of finite type either admits a dominated splitting or can be $\mathcal{C}^0$-approximated by one that \emph{stably} supports such measures. The stability holds in H\"older topology which is optimal. Additionally, for non-isometric cocycles, we show that the topological entropy of these bounded orbits is strictly less than that of the base subshift.

math.DS↗

arXiv2024

Cocycles with Quasi-Conformality I: Stability and abundance

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.

math.DS↗