In this paper, we study the effective dimension of points in infinite fractal trees generated recursively by a finite tree over some alphabet. Using unequal costs coding, we associate a length function with each such fractal tree and show that the channel capacity of the length function is equal to the similarity dimension of the fractal tree (up to a multiplicative constant determined by the size of the alphabet over which our tree is defined). Using this result, we derive formulas for calculating the effective dimension and strong effective dimension of points in fractal trees, establishing analogues of several results due to Lutz and Mayordomo, who studied the effective dimension of points in self-similar fractals in Euclidean space. Lastly, we explore the connections between the channel capacity of a length function derived from a finite tree and the measure of maximum entropy on a related directed multigraph that encodes the structure of our tree, drawing on work by Abram and Lagarias on path sets, where a path set is a generalization of the notion of a sofic shift.

翻译：本文研究了由有限树在某个字母表上递归生成的无限分形树中点的有效维数。利用不等代价编码，我们将每个此类分形树与一个长度函数相关联，并证明该长度函数的信道容量等于分形树的相似维数（相差一个由定义树的字母表大小决定的乘法常数）。基于这一结果，我们推导出计算分形树中点有效维数和强有效维数的公式，建立了与Lutz和Mayordomo关于欧几里得空间中自相似分形点有效维数的若干结果的对应。最后，我们探讨了由有限树导出的长度函数的信道容量与编码树结构的有向多重图上最大熵测度之间的联系，这一部分借鉴了Abram和Lagarias在路径集（sofic移位概念的推广）方面的研究成果。