Mean Hausdorff Dimension of Some Infinite Dimensional Fractals for Amenable Group Actions

TL;DR

This paper calculates the mean Hausdorff dimensions of various infinite dimensional fractals under amenable group actions, expanding understanding of fractal geometry in infinite-dimensional spaces.

Contribution

It introduces formulas for mean Hausdorff dimensions of self-similar, homogeneous, and carpet fractals in infinite dimensions for amenable group actions.

Findings

Mean Hausdorff dimensions are explicitly calculated for three types of infinite dimensional fractals.

Results extend fractal dimension theory to infinite-dimensional settings.

Provides new insights into the structure of fractals under group actions.

Abstract

For the countable discrete amenable group actions, we calculate the mean Hausdorff dimensions of three types of infinite dimensional fractal systems, the self-similar systems, homogeneous systems in the infinite-dimensional torus, and the infinite dimensional-carpets.