Multifractal formalism for non-selfsimilar functions
Journal Article
رابط المنشور على الويب:
المجلة \ الصحيفة:
Integral Transforms and Special Functions
رقم العدد:
15(2004), 189-207
الصفحات:
189-207
مستخلص المنشور:
A function f is self-similar if, modulo an error regular function g, it (i.e., f ) is invariant under specific transformations involving mainly dilations and translations (known as the piecewise linear dynamical systems). The multi-fractal formalism for functions has been proved to be valid for a large class of self-similar functions. For the computation of the spectrum of singularities for such functions, Gibbs measures were constructed. The authors have extended the validity for quasi-self-similar functions of the form (17). The authors also gave some examples of quasi-self-similar functions that are not self-similar.
In this paper we will study the multifractal properties for another class of non-self-similar (nor quasi-self-similar) functions: the sum F = F 1 + F 2 of self-similar functions. For such sums, the construction of associated Gibbs measures is not possible. We use instead the Frostman method. We also compare the spectrum of F to the spectra of F 1 and F 2. All our results remain valid for more general sums corresponding to hyperbolic dynamical systems.