Higher-dimensional multifractal analysis

Multifractal and higher dimensional zeta functions

In this paper, we generalize the zeta function for a fractal string (as in [18]) in several directions. We first modify the zeta function to be associated with a sequence of covers instead of the usual definition involving gap lengths. This modified zeta function allows us to define both a multifractal zeta function and a zeta function for higher-dimensional fractal sets. In the multifractal ca...

Multiscale multifractal detrended-fluctuation analysis of two-dimensional surfaces.

Two-dimensional (2D) multifractal detrended fluctuation analysis (MF-DFA) has been used to study monofractality and multifractality on 2D surfaces, but when it is used to calculate the generalized Hurst exponent in a fixed time scale, the presence of crossovers can bias the outcome. To solve this problem, multiscale multifractal analysis (MMA) was recent employed in a one-dimensional case. MMA ...

Multifractal analysis for all

Multifractal formalism and analysis have been described many times (Lopes and Betrouni, 2009). There are several approaches to the implementation of multifractal analysis and there numerous ways to present these (Olsen, 1995; Wendt et al., 2009; Ihlen, 2012). Ihlen’s article, in the June 4th 2012 issue of Frontiers of Physiology, entitled “Introduction to multifractal detrended fluctuation anal...

Box-counting multifractal analysis.

Multifractal Structure of Two-dimensional Horseshoes

We give a complete description of the dimension spectra of Birkhoff averages on a hyperbolic set of a surface diffeomorphism. The main novelty is that we are able to consider simultaneously Birkhoff averages into the future and into the past, i.e., both for positive and negative time. We emphasize that the description of these spectra is not a consequence of the available results in the case of...