Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|^(-1), and that the set of t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega.Comment: latex, 36 pages, no figures, to appear on Journ. Funct. Analysi

The Fractal Geometry of the Cosmic Web and its Formation

The cosmic web structure is studied with the concepts and methods of fractal geometry, employing the adhesion model of cosmological dynamics as a basic reference. The structures of matter clusters and cosmic voids in cosmological N-body simulations or the Sloan Digital Sky Survey are elucidated by means of multifractal geometry. A non-lacunar multifractal geometry can encompass three fundamental descriptions of the cosmic structure, namely, the web structure, hierarchical clustering, and halo distributions. Furthermore, it explains our present knowledge of cosmic voids. In this way, a unified theory of the large-scale structure of the universe seems to emerge. The multifractal spectrum that we obtain significantly differs from the one of the adhesion model and conforms better to the laws of gravity. The formation of the cosmic web is best modeled as a type of turbulent dynamics, generalizing the known methods of Burgers turbulence.Comment: 35 pages, 8 figures; corrected typos, added references; further discussion of cosmic voids; accepted by Advances in Astronom

Dimensions and singular traces for spectral triples, with applications to fractals

Given a spectral triple (A,D,H), the functionals on A of the form a -> tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and omega is a generalised limit. When tau_omega is the Dixmier trace, the unique exponent d giving rise possibly to a non-trivial functional is called Hausdorff dimension, and the corresponding functional the (d-dimensional) Hausdorff functional. It is shown that the Hausdorff dimension d coincides with the abscissa of convergence of the zeta function of |D|^(-1), and that the set of t's for which there exists a singular trace tau_omega giving rise to a non-trivial functional is an interval containing d. Moreover, the endpoints of such traceability interval have a dimensional interpretation. The corresponding functionals are called Hausdorff-Besicovitch functionals. These definitions are tested on fractals in R, by computing the mentioned quantities and showing in many cases their correspondence with classical objects. In particular, for self-similar fractals the traceability interval consists only of the Hausdorff dimension, and the corresponding Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More generally, for any limit fractal, the described functionals do not depend on the generalized limit omega.Comment: latex, 36 pages, no figures, to appear on Journ. Funct. Analysi

Certain singular distributions and fractals

In the present article, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.Comment: 22 pages. This research is an extension of investigation arXiv:1808.00395. arXiv admin note: text overlap with arXiv:1706.0154

Multifractal concentrations of inertial particles in smooth random flows

Collisionless suspensions of inertial particles (finite-size impurities) are studied in 2D and 3D spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterise in full generality the mechanisms leading to the formation of strong inhomogeneities in the particle concentration. Phenomenological arguments are used to show that in 2D, heavy particles form dynamical fractal clusters when their Stokes number (non-dimensional viscous friction time) is below some critical value. Numerical simulations provide strong evidence for this threshold in both 2D and 3D and for particles not only heavier but also lighter than the carrier fluid. In 2D, light particles are found to cluster at discrete (time-dependent) positions and velocities in some range of the dynamical parameters (the Stokes number and the mass density ratio between fluid and particles). This regime is absent in 3D, where evidence is that the Hausdorff dimension of clusters in phase space (position-velocity) remains always above two. After relaxation of transients, the phase-space density of particles becomes a singular random measure with non-trivial multiscaling properties. Theoretical results about the projection of fractal sets are used to relate the distribution in phase space to the distribution of the particle positions. Multifractality in phase space implies also multiscaling of the spatial distribution of the mass of particles. Two-dimensional simulations, using simple random flows and heavy particles, allow the accurate determination of the scaling exponents: anomalous deviations from self-similar scaling are already observed for Stokes numbers as small as $10^{-4}$.Comment: 21 pages, 13 figure

On the fractal characteristics of a stabilised Newton method

In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed