The relative Renyi dimention

N. Yu. Svetova

Problemy analiza, Tome 1 (2012) no. 1, pp. 15-23

Résumé

Recently, many authors are discussing the use of methods of fractal geometry [5] to compare the distributions of the various measures. However, in practical applications, comparison of distributions by comparing the calculated multifractal spectra can be difficult. It often happens that a completely different distributions of measures can give very imperceptible differences in the spectra. To solve this problem, some authors [4,10] propose to use different methods of direct comparison of distributions. These methods are generalizations of the classical multifractal analysis developed in the works L. Olsen [9], K.-S. Lo and S.-M. Ngai [8] and others. Based on the idea of multifractal analysis [9] and the mutual multifractal analysis [1,2] we propose to introduce new concepts of relative Renyi dimensions for coverings, packings and partitions, as well as we establish some connection between them. It should be noted that these dimension proved mathematically rigorous new analogues «new relative multifractal spectrum of dimensions» proposed for purely practical purposes, R. Dansereau and W. Kinser [6].