A multifractal formalism for new general fractal measures

In this study, we will introduce an innovative and comprehensive multifractal framework, substantiating counterparts to the classical findings in multifractal analysis and We embark on an exploration of the mutual singularity existing between the broader multifractal Hausdorff and packing measures within an expansive framework. An exemplar of this framework involves the application of the ”second-order” multifractal formalism to our core results, elucidating a realm of compact subsets within a self-similar fractal structure — a familiar illustration of an infinite-dimensional metric space. Within this context, we provide estimations for both the overall Hausdorff and packing dimensions. It is noteworthy that these outcomes offer novel validations for theorems underpinning the multifractal formalism, rooted in these comprehensive multifractal measures. Furthermore, these findings remain valid even at points q where the multifractal functions governing Hausdorff and packing dimensions diverge. Additionally, we introduce the concepts of lower and upper relative multifractal box dimensions, accompanied by the general Rényi dimensions. A comparison is drawn between these dimensions and the general multifractal Hausdorff dimension, along with the general multifractal pre-packing dimension. Finally, we establish density conclusions pertaining to the multifractal extension of the centered Hausdorff and packing measures. Specifically, we unveil a decomposition theorem akin to Besicovitch’s theorem for these measures. This theorem facilitates a division into two components – one characterized as regular and the other as irregular – thus enabling a targeted analysis of each segment. Following this dissection, we seamlessly reintegrate these segments while preserving their intrinsic density attributes. The impetus behind investigating these comprehensive measures arises when a set E holds either a conventional Hausdorff measure of zero or infinity; In such cases, we can identify a function ϕs that bestows a positive and finite general Hausdorff measure upon the set E, and studies the dimensions of infinitely dimensional sets.