Uniform quasi-multiplicativity of locally constant cocycles and applications (Q6547200)

The setting here is a \(\mathrm{GL}_d(\mathbb{R})\)-valued cocycle \(A\) over a one-sided full shift, and the main result is that irreducibility of \(A^t\) for every \(t\vert d\) and of the exterior product cocycles \(A^{\wedge m}\) for \(1\leqslant m\leqslant d-1\) implies that the cocycle has a spanning property and hence satisfies a quasi-multiplicative property. The latter property is used to deduce \(\psi\)-mixing for the unique subadditive equilibrium Gibbs state and to calculate the Hausdorff dimension of shrinking target sets and recurrence sets of relevance to problems in Diophantine number theory.