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 $e^{\hat{I}{»}_{1}n}\ge e^{\hat{I}{»}_{2}n}\ge \dots \ge e^{\hat{I}{»}_{k}n}\ge e^{\hat{I}{»}_{k+1}n}$ at every periodic point of period $n$, and $\hat{I}{»}_k>\hat{I}{»}_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if $\hat{I}{»}_k>0>\hat{I}{»}_{k+1}$ 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.

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