Multifractal Spectrum and Thermodynamical Formalism of the Farey Tree

The task of comparing the Hausdorff spectrum, the computational spectrum, and the Legendre spectrum of a fractal set endowed with a probability measure, was tackled by several authors - Cawley and Mauldin, Riedi and Mandelbrot, among others. For self-similar measures all three spectra coincide. We compare these spectra for the hyperbolic measure (inducing the Farey Tree partition), fundamentally different from the self-similar one.Comment: 35 pages, 4 figure

Multifractal analysis of a road-to-crisis in a Faraday experiment

A thin layer of liquid in a horizontal cell is subjected to a periodic vertical force with two control parameters: acceleration and frequency. The influence of the rheological behavior of the fluid was considered over the empirically obtained results, which were subjected to a multfractal spectrum process. As control parameters varied, so did the corresponding spectra, according to theoretical models; each change of state was visually detected, which permitted interpreting the corresponding spectral changes

A formula for the fractal dimension d approx. 0.87 of the Cantorian set underlying the Devil's staircase associated with the Circle Map

The Cantor set complementary to the Devil's Staircase associated with the Circle Map has a fractal dimension d approximately equal to 0.87, a value that is universal for a wide range of maps, such results being of a numerical character. In this paper we deduce a formula for such dimensional value. The Devil's Staircase associated with the Circle Map is a function that transforms horizontal unit interval I onto vertical I, and is endowed with the Farey-Brocot (F-B) structure in the vertical axis via the rational heights of stability intervals. The underlying Cantor-dust fractal set Omega in the horizontal axis --Omega contained in I, with fractal dimension d(Omega) approx. 0.87-- has a natural covering with segments that also follow the F-B hierarchy: therefore, the staircase associates vertical I (of unit dimension) with horizontal Omega in I (of dimension approx. 0.87), i.e. it selects a certain subset Omega of I, both sets F- B structured, the selected Omega with smaller dimension than that of I. Hence, the structure of the staircase mirrors the F- B hierarchy. In this paper we consider the subset Omega-F-B of I that concentrates the measure induced by the F-B partition and calculate its Hausdorff dimension, i.e. the entropic or information dimension of the F-B measure, and show that it coincides with d(Omega) approx. 0.87. Hence, this dimensional value stems from the F-B structure, and we draw conclusions and conjectures from this fact. Finally, we calculate the statistical "Euclidean" dimension (based on the ordinary Lebesgue measure) of the F-B partition, and we show that it is the same as d(Omega-F-B), which permits conjecturing on the universality of the dimensional value d approximately equal to 0.87.Comment: 53 pages double spaced, 1 figur

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials $\Phi$ taking values in $\R^{\N}$, we give a formula for the Hausdorff dimension of $J_\Phi(\alpha)$, the $\alpha$-level set of the Birkhoff average, on a typical infinitely generated self-affine set. We also show that for bounded potentials $\Phi$, the Hausdorff dimension of $J_\Phi(\alpha)$ is given by the maximum of the critical value for the pressure and the supremum of Lyapunov dimensions of invariant measures $\mu$ for which $\int\Phi\,d\mu=\alpha$. Our multifractal results are new in both the finitely generated and the infinitely generated setting

Geometry of fractional spaces

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.Comment: 90 pages, 6 figures, 4 tables. v2: section 5 revised, result unchanged; v3: minor typos correcte