可数生成自仿射集的热力学形式
Thermodynamic formalism of countably generated self-affine sets
AI总结：
本文发展可数仿射迭代函数系统的热力学形式，证明其平衡态存在性并推广有限情形的刻画，还得到可数仿射迭代函数系统仿射维数等于有限子系统仿射维数上确界等相关维数推论。
摘要：
本文通过证明平衡态的存在性，并将有限仿射迭代函数系统平衡态的已有刻画推广至可数无穷情形，进一步发展了含可数多个变换的仿射迭代函数系统的热力学形式。作为应用，在温和条件下，我们证明了可数仿射迭代函数系统的仿射维数等于其有限子系统仿射维数的上确界。我们推导了关于1、2、3维下满足温和确定性假设、以及任意维数下具 generic 平移的可数生成自仿射集的 Hausdorff 维数的推论。
英文摘要：
In this article, we further develop the thermodynamic formalism of affine iterated function systems with countably many transformations by showing the existence and extending earlier characterisations of the equilibrium states of finite affine iterated function systems to the countably infinite case. As an application, under mild conditions, we prove that the affinity dimension of a countable affine iterated function system is equal to the supremum of the affinity dimensions of its finite subsystems. We deduce corollaries concerning the Hausdorff dimension of countably generated self-affine sets in dimensions $1$, $2$, and $3$ satisfying mild deterministic assumptions and in arbitrary dimension with generic translations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
