Intermediate dimensions
| العنوان: | Intermediate dimensions |
|---|---|
| المؤلفون: | Kenneth J. Falconer, Jonathan M. Fraser, Tom Kempton |
| المساهمون: | The Leverhulme Trust, EPSRC, University of St Andrews. Pure Mathematics |
| المصدر: | Falconer, K J, Fraser, J M & Kempton, T 2019, ' Intermediate dimensions ', Mathematische Zeitschrift . https://doi.org/10.1007/s00209-019-02452-0 |
| سنة النشر: | 2018 |
| مصطلحات موضوعية: | primary: 28A80, secondary: 37C45, Mathematics(all), Box dimension, General Mathematics, T-NDAS, 010102 general mathematics, Hausdorff dimension, Metric Geometry (math.MG), Dynamical Systems (math.DS), 01 natural sciences, 010101 applied mathematics, Mathematics - Metric Geometry, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Self-affine carpet, QA Mathematics, 0101 mathematics, Mathematics - Dynamical Systems, QA |
| الوصف: | We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U| \leq |V|^\theta$ for all sets $U, V$ used in a particular cover, where $\theta \in [0,1]$ is a parameter. Thus, when $\theta=1$ only covers using sets of the same size are allowable, and we recover the box dimensions, and when $\theta=0$ there are no restrictions, and we recover Hausdorff dimension. We investigate many properties of the intermediate dimension (as a function of $\theta$), including proving that it is continuous on $(0,1]$ but not necessarily continuous at $0$, as well as establishing appropriate analogues of the mass distribution principle, Frostman's lemma, and the dimension formulae for products. We also compute, or estimate, the intermediate dimensions of some familiar sets, including sequences formed by negative powers of integers, and Bedford-McMullen carpets. Comment: 19 pages, 1 figure. To appear in Mathematische Zeitschrift |
| وصف الملف: | application/pdf |
| اللغة: | English |
| URL الوصول: | https://explore.openaire.eu/search/publication?articleId=doi_dedup___::aaeeff3329af34bff264265c9df32b1b http://arxiv.org/abs/1811.06493 |
| حقوق: | OPEN |
| رقم الأكسشن: | edsair.doi.dedup.....aaeeff3329af34bff264265c9df32b1b |
| قاعدة البيانات: | OpenAIRE |
| الوصف غير متاح. |