关于自相似分形乘积上的水平集问题
On the horizon problem over the product of self-similar fractals
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中文总结 AI 辅助
本文研究满足开集条件的自相似分形乘积上连续函数图像的盒维数，证明存在满足水平性质的普遍曲面，其盒维数与水平集盒维数的差值为分形的盒维数。
中文摘要 AI 辅助
本文首先研究定义在两个自相似分形（记为𝔉）乘积上的连续函数图像的盒维数结果，其中分形𝔉满足开集条件。随后，我们证明存在一个普遍曲面，该曲面由定义在两个𝔉乘积上的连续函数确定，满足水平性质（与盒维数相关），即该普遍曲面的盒维数大于其水平集的盒维数，差值为常数η，其中η是自相似分形𝔉的盒维数。
英文摘要
In this article, we first investigate the box dimension results for the graphs of continuous functions defined on the product of two self-similar fractals ({\mathfrak{F}}s), where the fractal {\mathfrak{F}} satisfies the open set condition. Following this, we manifest the existence of a prevalent surface, defined by a continuous function over the product of two {\mathfrak{F}}s, that satisfies the horizon property (associated with the box dimension), i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant η, where η is the box dimension of the self-similar fractal {\mathfrak{F}}.
