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Uniform quasi-multiplicativity of locally constant cocycles and applications

Volume 275 / 2024

Reza Mohammadpour, Kiho Park Studia Mathematica 275 (2024), 85-98 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.

Authors

  • Reza MohammadpourDepartment of Mathematics
    Uppsala University
    SE-75106 Uppsala, Sweden
    ORCID: 0000-0003-3999-8114
    e-mail
  • Kiho ParkSchool of Mathematics
    KIAS
    Seoul, 02455, Republic of Korea
    e-mail

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