Generalized fractal dimensions: equivalences and basic properties

AbstractGiven a positive probability Borel measure μ on R, we establish some basic properties of the associated functions τμ±(q) and of the generalized fractal dimensions Dμ±(q) for q∈R. We first give the equivalence of the Hentschel–Procaccia dimensions with the Rényi dimensions and the mean-q dimensions, for q>0. We then use these relations to prove some regularity properties for τμ±(q) and Dμ±(q); we also provide some estimates for these functions, in particular estimates on their behaviour at ±∞, as well as for the dimensions corresponding to convolution of two measures. We finally present some calculations for specific examples illustrating the different cases met in the article