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<p>Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry.<OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> <p>The calculation of Hausdorff dimension of homogeneous sets in the circle.</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> <p>The coincidence of Hausdorff and Minkowski dimensions for self-similar sets.</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(3)</ItemNumber> <ItemContent> <p>The calculation of Hausdorff dimension of Bedford–McMullen carpets.</p> </ItemContent> </ListItem> </OrderedList></p><p>We develop their analogues for mean Hausdorff dimension:<UnorderedList Mark="None"> <ItemContent> <p>(1′) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite-dimensional torus.</p> </ItemContent> <ItemContent> <p>(2′) The coincidence of mean Hausdorff dimension and metric mean dimension for self-similar systems.</p> </ItemContent> <ItemContent> <p>(3′) The calculation of mean Hausdorff dimension of infinite-dimensional carpets.</p> </ItemContent> </UnorderedList></p>

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Mean Hausdorff dimension of some infinite-dimensional fractals

  • 期刊论文

  • 发表日期：

    2025年1月21日

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  • Masaki Tsukamoto

摘要

Mean Hausdorff dimension is a dynamical version of Hausdorff dimension. It provides a way to dynamicalize geometric measure theory. We pick up the following three classical results of fractal geometry. (1)

The calculation of Hausdorff dimension of homogeneous sets in the circle.

(2)

The coincidence of Hausdorff and Minkowski dimensions for self-similar sets.

 (3)

The calculation of Hausdorff dimension of Bedford–McMullen carpets.

We develop their analogues for mean Hausdorff dimension:

(1′) The calculation of mean Hausdorff dimension of homogeneous systems in the infinite-dimensional torus.

(2′) The coincidence of mean Hausdorff dimension and metric mean dimension for self-similar systems.

(3′) The calculation of mean Hausdorff dimension of infinite-dimensional carpets.