Continuous SL(d,R) cocycles over positive-entropy subshifts of finite type either admit a dominated splitting or can be C0-approximated to C^α-stably support positive-entropy ergodic measures on bounded orbits.
Yoccoz, Some questions and remarks about \(SL(2, R)\) cocycles , Modern dynamical systems and applications
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Cocycles with Quasi-Conformality II: Ergodic measures with positive entropy
Continuous SL(d,R) cocycles over positive-entropy subshifts of finite type either admit a dominated splitting or can be C0-approximated to C^α-stably support positive-entropy ergodic measures on bounded orbits.