Uniform quasi-multiplicativity of locally constant cocycles and applications
Tom 275 / 2024
Streszczenie
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.