Dominated splittings and periodic data for quasi-compact operator cocycles

Abstract

For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that if the moduli of the $(k+1)$-largest eigenvalues of the cocycle are $e^{\lambda_1n}\geq e^{\lambda_2n}\geq \ldots\geq e^{\lambda_kn}\geq e^{\lambda_{k+1}n}$ at every periodic point of period $n$, and $\lambda_k>\lambda_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if $\lambda_k>0>\lambda_{k+1}$ then the cocycle is uniformly hyperbolic. Furthermore, we are able to obtain these same conclusions even when the eigenvalues are only close to constant, not strictly constant.