I am currently reading the following paper by F.Takens:
Multifractal analysis of dimensions and entropies.
This paper discusses two different measures. One is generalized entropies and the other is generalized dimensions, however as far as I can see it doesn't discuss any association between the two. Can somebody please in a qualitative or [even better] quantitative give me an explanation about the relation between the two?
(Assuming limits exist and upper and lower quantities are equal to simplify equations)
Given a partition $p$ enumerated by the index $i$, the Rényi information is defined as (Rényi 1960, 1970),
$$I_q(p) \equiv \frac{1}{q-1}\log\sum_{i}p_i^\beta$$
and not with $1/(1-q)$ as presented in the paper (presumably a typo in the sentence
is replaced by a more general Rényi’s information [...][Rényi’s entropy?])The Rényi dimensions are defined as (Mandelbrot 1974; Hentschel and Procaccia 1983; Grassberger 1983)
$$ D_q (p) \equiv \lim_{\varepsilon\rightarrow 0} \frac{I_q(p)}{\log \varepsilon}$$
(the relation you are searching?) which can be rewritten as the paper presents:
$$ D_q (p) = \lim_{\varepsilon\rightarrow 0} \frac{1}{q-1}\frac{\log \sum_i p_i^q}{\log \varepsilon}$$
Qualitatively, one reason why dimensions are introduced here is that entropy (or minus information) scales with increasing $\varepsilon$. It is thus natural to define an $\epsilon$-independent quantity to generalize the topological dimension ($D_0$) and information dimension ($D_1$) for arbitrary $q$.
There are deeper reasons why they must be related; I suggest the book by Beck and Schlogl "Thermodynamics of chaotic systems - an introduction" for that.