Different types of multifractal measures in separable metric spaces and their applications
The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the p...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023650?viewType=HTML |
| _version_ | 1797849797062819840 |
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| author | Najmeddine Attia Bilel Selmi |
| author_facet | Najmeddine Attia Bilel Selmi |
| author_sort | Najmeddine Attia |
| collection | DOAJ |
| description | The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context. |
| first_indexed | 2024-04-09T18:49:57Z |
| format | Article |
| id | doaj.art-d3b60e06f48e4676b76091e6ce5ee7ac |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-09T18:49:57Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-d3b60e06f48e4676b76091e6ce5ee7ac2023-04-10T01:37:36ZengAIMS PressAIMS Mathematics2473-69882023-03-0186128891292110.3934/math.2023650Different types of multifractal measures in separable metric spaces and their applicationsNajmeddine Attia0Bilel Selmi 11. Department of Mathematics and Statistics, College of Science, King Faisal University, PO. Box : 400 Al-Ahsa 31982, Saudi Arabia2. Analysis, Probability and Fractals Laboratory LR18ES17, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, 5000-Monastir, TunisiaThe properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and the relative packing measures $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that $ {\mathscr H}_{ {\boldsymbol{\mu}}}^{q, t} $ and $ {\mathscr P}_{ {\boldsymbol{\mu}}}^{q, t} $ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.https://www.aimspress.com/article/doi/10.3934/math.2023650?viewType=HTMLgeneralized hausdorff measuregeneralized packing measurehenstock-thomson variationregularitiesdensitiesdoubling measures |
| spellingShingle | Najmeddine Attia Bilel Selmi Different types of multifractal measures in separable metric spaces and their applications AIMS Mathematics generalized hausdorff measure generalized packing measure henstock-thomson variation regularities densities doubling measures |
| title | Different types of multifractal measures in separable metric spaces and their applications |
| title_full | Different types of multifractal measures in separable metric spaces and their applications |
| title_fullStr | Different types of multifractal measures in separable metric spaces and their applications |
| title_full_unstemmed | Different types of multifractal measures in separable metric spaces and their applications |
| title_short | Different types of multifractal measures in separable metric spaces and their applications |
| title_sort | different types of multifractal measures in separable metric spaces and their applications |
| topic | generalized hausdorff measure generalized packing measure henstock-thomson variation regularities densities doubling measures |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023650?viewType=HTML |
| work_keys_str_mv | AT najmeddineattia differenttypesofmultifractalmeasuresinseparablemetricspacesandtheirapplications AT bilelselmi differenttypesofmultifractalmeasuresinseparablemetricspacesandtheirapplications |