标题： 关于自共形集上的Gibbs测度的指数混合及其应用 显示英文标题

标题： Exponential mixing for Gibbs measures on self-conformal sets and applications

摘要： 本文中，我们证明了：若由满足开集条件的 $C^{1+\alpha}$ 自相似迭代函数系（IFS）生成的 $\mathbb{R}^d$ 上的 Gibbs 测度是指数混合的。 我们利用这一结果得到了与任意此类集合相关的回归集和递减目标子集的本质上最优的渐近计数陈述。 特别是，我们给出了动力系统的一些显式例子，其中回归集表现出（出乎意料的）行为，在递减目标设定中不存在这种行为。 在建立我们的指数混合结果的过程中，我们将 Mattila 的自相似集刚性定理扩展到了没有分离条件的自共形集，并且适用于任意 Gibbs 测度。 显示英文摘要

摘要： In this paper, we show that Gibbs measures on self-conformal sets generated by a $C^{1+\alpha}$ conformal IFS on $\mathbb{R}^d$ satisfying the OSC are exponentially mixing. We exploit this to obtain essentially sharp asymptotic counting statements for the recurrent and the shrinking target subsets associated with any such set. In particular, we provide explicit examples of dynamical systems for which the recurrent sets exhibit (unexpected) behaviour that is not present in the shrinking target setup. In the process of establishing our exponential mixing result we extend Mattila's rigidity theorem for self-similar sets to self-conformal sets without any separation condition and for arbitrary Gibbs measures.