Periodic approximation of Lyapunov exponents for cocycles admitting invariant holonomies

Abstract

Classical results establish that the Lyapunov exponents of an ergodic measure for linear cocycles over hyperbolic systems can be approximated by the Lyapunov exponents of periodic orbits, provided the cocycle is Hölder continuous. A recent counterexample by Bochi demonstrates that this approximation property fails in general if the Hölder assumption is relaxed to mere continuity. In this paper, we introduce a geometric condition that successfully substitutes this analytical regularity hypothesis. More precisely, we prove that if a cocycle - even a discontinuous one - admits a continuous family of invariant holonomies, the periodic approximation of Lyapunov exponents still holds. Our geometric approach yields a proof that is substantially simpler and more direct than existing arguments in the literature, even when applied to classical settings such as fiber-bunched cocycles for which previous results were already available.