The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917014041264128 |
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| author | Anttila, Riku Eriksson-Bique, Sylvester |
| author_facet | Anttila, Riku Eriksson-Bique, Sylvester |
| contents | This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon--Kleiner and the super-multiplicativity inequalities. The latter are established on Menger sponges for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15692 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces Anttila, Riku Eriksson-Bique, Sylvester Metric Geometry Analysis of PDEs 30L10, 20F65, 51F99, 53C23, 28A78 This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carpet and the higher dimensional Menger sponges, but also a rich class of new examples. The main result of the work roughly speaking states: If the construction is sufficiently symmetric then the limiting object supports useful moduli estimates, namely the Combinatorial Loewner property of Bourdon--Kleiner and the super-multiplicativity inequalities. The latter are established on Menger sponges for which it had not been previously known. The main new technique the work offers is a general framework of flows and resistance estimates. |
| title | The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces |
| topic | Metric Geometry Analysis of PDEs 30L10, 20F65, 51F99, 53C23, 28A78 |
| url | https://arxiv.org/abs/2408.15692 |