标题： 平衡状态的统计性质及其在纤维束矩阵胞腔中的应用 显示英文标题

标题： Statistical properties of equilibrium states for fiber-bunched matrix cocycles and applications

摘要： 我们对霍尔德连续纤维紧致矩阵cocycles、阿诺索夫微分同胚和双曲排斥子的热力学形式主义做出贡献。 具体来说，我们证明了在拓扑混合的有限型子移位上，$1$类型的纤维紧致cocycles$\mathcal{A}$在参数范围$t \in (-t_*, +\infty)$内与非加性势能族$\{t \log \|\mathcal{A}^n\|\}_{n \in \mathbb{N}}$相关的唯一吉布斯平衡态$\mu_t$存在，其中$t_* > 0$。 此外，这些平衡态是$\psi$-混合的，因此是弱伯努利的。 此外，这些结果使我们能够推导出双曲排斥子和阿诺索夫微分同胚的开集的热力学形式主义的相关结论。 特别是，它对Gatzouras和Peres提出的关于$C^1$-开集的$\alpha$-纤维紧致双曲排斥子的猜想给出了肯定的回答。 显示英文摘要

摘要： We contribute to the thermodynamic formalism of H\"older continuous fiber-bunched matrix cocycles, Anosov diffeomorphisms, and hyperbolic repellers. Specifically, we prove that $1$-typical fiber-bunched cocycles $\mathcal{A}$ over topologically mixing subshifts of finite type admit a unique Gibbs equilibrium state $\mu_t$ associated with the non-additive family of potentials $\{t \log \|\mathcal{A}^n\|\}_{n \in \mathbb{N}}$, for a range of parameters $t \in (-t_*, +\infty)$, where $t_* > 0$. Furthermore, these equilibrium states are $\psi$-mixing, therefore weak Bernoulli. In addition, these results allow us to derive consequences for the thermodynamic formalism of open sets of hyperbolic repellers and Anosov diffeomorphisms. In particular, it provides a positive answer to a conjecture posed by Gatzouras and Peres for $C^1$-open sets of $\alpha$-fiber-bunched hyperbolic repellers.