Uniform quasi-multiplicativity of locally constant cocycles and applications
Studia Mathematica
MSC: Primary 37D35; Secondary 37H15, 28A80, 37A44
DOI: 10.4064/sm230626-7-2
Published online: 24 April 2024
Abstract
We show that every locally constant cocycle $\mathcal A$ is $k$-quasi-multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal A^t$ and $\mathcal A^{\wedge m}$ are irreducible for every $t \,|\,d$ and $1\leq m \leq d-1$, then $\mathcal A$ is $k$-uniformly spannable for some $k\in \mathbb N$, which implies that $\mathcal A$ is $k$-quasi-multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $\psi $-mixing and calculate the Hausdorff dimension of cylindrical shrinking target sets and recurrence sets.
