470 research outputs found
Sixty Years of Fractal Projections
Sixty years ago, John Marstrand published a paper which, among other things,
relates the Hausdorff dimension of a plane set to the dimensions of its
orthogonal projections onto lines. For many years, the paper attracted very
little attention. However, over the past 30 years, Marstrand's projection
theorems have become the prototype for many results in fractal geometry with
numerous variants and applications and they continue to motivate leading
research.Comment: Submitted to proceedings of Fractals and Stochastics
Random walk through fractal environments
We analyze random walk through fractal environments, embedded in
3-dimensional, permeable space. Particles travel freely and are scattered off
into random directions when they hit the fractal. The statistical distribution
of the flight increments (i.e. of the displacements between two consecutive
hittings) is analytically derived from a common, practical definition of
fractal dimension, and it turns out to approximate quite well a power-law in
the case where the dimension D of the fractal is less than 2, there is though
always a finite rate of unaffected escape. Random walks through fractal sets
with D less or equal 2 can thus be considered as defective Levy walks. The
distribution of jump increments for D > 2 is decaying exponentially. The
diffusive behavior of the random walk is analyzed in the frame of continuous
time random walk, which we generalize to include the case of defective
distributions of walk-increments. It is shown that the particles undergo
anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the
finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced
though for small and intermediate times. In particular, it follows that
fractals generated by a particular class of self-organized criticality (SOC)
models give rise to enhanced diffusion. The analytical results are illustrated
by Monte-Carlo simulations.Comment: 22 pages, 16 figures; in press at Phys. Rev. E, 200
Harmonic analysis of iterated function systems with overlap
In this paper we extend previous work on IFSs without overlap. Our method
involves systems of operators generalizing the more familiar Cuntz relations
from operator algebra theory, and from subband filter operators in signal
processing.Comment: 37 page
Generalised dimensions of measures on almost self-affine sets
We establish a generic formula for the generalised q-dimensions of measures
supported by almost self-affine sets, for all q>1. These q-dimensions may
exhibit phase transitions as q varies. We first consider general measures and
then specialise to Bernoulli and Gibbs measures. Our method involves estimating
expectations of moment expressions in terms of `multienergy' integrals which we
then bound using induction on families of trees
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
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