Variational Principles for General Fractal Dimensions

The objective of this research is to establish a representation of the general Hausdorff and packing dimensions of compact sets in Euclidean space. This representation is formulated in terms of the lower and upper local dimensions with respect to two functions, denoted as h and g, which are associated with a probability measure defined on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>n$$\end{document}. Additionally, we establish a relationship that enables us to determine the general fractal dimensions by utilizing corresponding versions of these measures. These dimensions are defined as the supremum over the lower and upper general dimensions across all Borel probability measures. As a practical application of these findings, we introduce a variational principle for these general fractal dimensions. This principle establishes an equality between a parameter that is inherently linked to a given space or mapping and the supremum of specific values associated with a class of probability measures supported by the set, denoted as A. Furthermore, we propose two additional dimensions: the general concentration dimension and the general thin dimension. The concentration dimension is defined through the utilization of the L & eacute;vy concentration function and offers a relatively straightforward calculation method. It is closely connected to the mass distribution principle and often simplifies the determination of the general Hausdorff dimension of sets. The thin dimension, on the other hand, is grounded in the concept of the thin function, which can be seen as an anti-concentration function in relation to the L & eacute;vy function. This thin dimension also exhibits connections with the general fractal dimension.