On certain families of planar patterns and fractals

This survey article is dedicated to some families of fractals that were introduced and studied during the last decade, more precisely, families of Sierpi\'nski carpets: limit net sets, generalised Sierpi\'nski carpets and labyrinth fractals. We give a unifying approach of these fractals and several of their topological and geometrical properties, by using the framework of planar patterns.Comment: survey article, 10 pages, 7 figure

The Maximum Entropy principle and the nature of fractals

We apply the Principle of Maximum Entropy to the study of a general class of deterministic fractal sets. The scaling laws peculiar to these objects are accounted for by means of a constraint concerning the average content of information in those patterns. This constraint allows for a new statistical characterization of fractal objects and fractal dimension.Comment: 7 pages, RevTex, includes 2 PS figure

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Fractals from genomes: exact solutions of a biology-inspired problem

This is a review of a set of recent papers with some new data added. After a brief biological introduction a visualization scheme of the string composition of long DNA sequences, in particular, of bacterial complete genomes, will be described. This scheme leads to a class of self-similar and self-overlapping fractals in the limit of infinitely long constotuent strings. The calculation of their exact dimensions and the counting of true and redundant avoided strings at different string lengths turn out to be one and the same problem. We give exact solution of the problem using two independent methods: the Goulden-Jackson cluster method in combinatorics and the method of formal language theory.Comment: 24 pages, LaTeX, 5 PostScript figures (two in color), psfi

Unbiased estimation of multi-fractal dimensions of finite data sets

We present a novel method for determining multi-fractal properties from experimental data. It is based on maximising the likelihood that the given finite data set comes from a particular set of parameters in a multi-parameter family of well known multi-fractals. By comparing characteristic correlations obtained from the original data with those that occur in artificially generated multi-fractals with the {\em same} number of data points, we expect that predicted multi-fractal properties are unbiased by the finiteness of the experimental data.Comment: LaTeX, 17 pages, figures encapsulated as picture environment

On the equality of Hausdorff and box counting dimensions

By viewing the covers of a fractal as a statistical mechanical system, the exact capacity of a multifractal is computed. The procedure can be extended to any multifractal described by a scaling function to show why the capacity and Hausdorff dimension are expected to be equal.Comment: CYCLER Paper 93mar001 Latex file with 3 PostScript figures (needs psfig.sty

Critical Behavior of the Ferromagnetic Ising Model on a Sierpinski Carpet: Monte Carlo Renormalization Group Study

We perform a Monte Carlo Renormalization Group analysis of the critical behavior of the ferromagnetic Ising model on a Sierpi\'nski fractal with Hausdorff dimension $d_f\simeq 1.8928$. This method is shown to be relevant to the calculation of the critical temperature $T_c$ and the magnetic eigen-exponent $y_h$ on such structures. On the other hand, scaling corrections hinder the calculation of the temperature eigen-exponent $y_t$. At last, the results are shown to be consistent with a finite size scaling analysis.Comment: 16 pages, 7 figure