ABSTRACT
Besicovitch and his school discovered cases where the Hausdorff–Besicovitch dimension takes the “entropy-like” form −∑ pj log pj, where the pj are probabilitylike. The center of the present paper is occupied by related expressions of the form −log pj, suitably normalized. Being entropylike but nonaveraged, they will be called “ELNA codimennions.” Other new concepts discussed here are those of “ELNA dimensional sequences,” and “ELNA Hölder.” While all the existing fractal dimensions involve sets or measures the ELNA dimensional sequences, ELNA dimensions, and ELNA Hölders yield a richer structure because they involve sequences of sets and measures, as well as their limits, which can be nonempty or empty. “Survival” occurs when the limit is nonempty; if so an ELNA dimension and ELNA Hölder are positive and difficult proofs show that the ELNA dimension’s value is typically identical to those of a Hausdorff–Besicovitch dimension and of other fractal dimensions. But the ELNA dimensional sequence brings important additional information. For example, in the case of multifractals characterized by a Hölder spectrum f(α), it defines useful approximating functions fε(α). “Extinction” means that the limit of the sequence is empty; the ELNA dimension is then negative, and it is shown that it fulfills a surprising and novel role: It manages to give straightforward interpretations and a numerical value to the so-far empty notion of “the degree of emptiness of an empty set.” One interpretation is geometric, in terms of the actual or formal embedding of the generating procedure in a higher dimensional space. The second interpretation is statistical, in terms of a novel procedure called “supersampling,” which is motivated by a novel “lateral” passage to the limit. The practical usefulness of negative-valued ELNA dimensions shows that the need may exist in physics for characteristics that become lost in asymptotic results (often described as “fine-grained thermodynamics”) but are present in preasymptotic results (“coarse-grained thermodynamics”) and can be usefully combined with lateral preasymptotics.
