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

TL;DR

This paper develops a principle to produce positive-entropy ergodic measures in linear cocycles over chaotic systems, and characterizes cocycles either admitting dominated splittings or approximable by such measures.

Contribution

It introduces a robust method for constructing positive-entropy measures and characterizes cocycles based on their splitting properties and approximability.

Findings

Established a multiple covering principle for positive-entropy measures.

Proved that cocycles either admit a dominated splitting or can be approximated by ones that support such measures.

Showed that for non-isometric cocycles, the entropy of bounded orbits is less than that of the base system.

Abstract

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 $C^0$-approximated by one that $C^\alpha$-stably supports such measures ($\alpha>0$). Additionally, for non-isometric cocycles, we show that the topological entropy of these bounded orbits is strictly less than that of the base subshift.