Uniform quasi-multiplicativity of locally constant cocycles and applications

Reza Mohammadpour; Kiho Park

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arXiv · 2209.08999

Uniform quasi-multiplicativity of locally constant cocycles and applications

Reza Mohammadpour·Kiho Park

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Abstract

In this paper, we show that a 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 $t \mid d$ 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 and recurrence sets.

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Reza Mohammadpour, Kiho Park. 2022-09-19. Uniform quasi-multiplicativity of locally constant cocycles and applications. https://arxiv.org/abs/2209.08999

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