Abstract:
A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be constructed by this method. These tilings can be used to extend a fractal transformation defined on the attractor of a contractive IFS to a fractal transformation on the entire space upon which the IFS acts.

Abstract:
We introduce a fractal version of the pinwheel substitution tiling. There are thirteen basic prototiles, all of which have fractal boundaries. These tiles, along with their reflections and rotations, create a tiling space which is mutually locally derivable from the pinwheel tiling space. Interesting rotational properties, symmetries, and relative tile frequency are discussed for the tiling space associated with the fractal pinwheel tiling.

Abstract:
In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For each initial tile choice the number of possible configurations is finite. This combinatorial problem recalls the famous Escher tiling problem [2]. By using the Burnside lemma we show that there are exactly 232 really different fractals when the initial tile is a particular 2x2 matrix. Partial results are also presented in the 3x3 case when the initial tile presents some symmetry properties.

Abstract:
We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between tiles in the tiling. We show that each spectral triple is $\theta$-summable and respects the hierarchy of the substitution system. To elucidate our results we construct a fractal tree on the Penrose tiling and explicitly show how it gives rise to a collection of spectral triples.

Abstract:
Geomagnetic field time variations can be analysed by means of fractal and chaotic techniques. This new kind of analysis can definitely help in studying geophysical signals even when their chaotic and/or fractal aspects are not so obvious. In this short paper some examples of this new kind of analysis are presented in the case of geomagnetic time variation data from L'Aquila Observatory.

Abstract:
In this article we will describe some types of retractions of chaotic manifold. We also will introduce a definition for the fractional dimension of the chaotic manifold. Some theorems related to them will be obtained. Some applications will also be mentioned in this paper.

Abstract:
Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis -- a widely used method in time series analysis -- and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that, in turn, qualitatively agree with the previous experimental data.

Abstract:
localized rain events have been found to follow power-law distributions over several decades, suggesting parallels between precipitation and seismic activity [o. peters et al., prl 88, 018701 (2002)]. similar power laws can be generated by treating raindrops as passive tracers advected by the velocity field of a two-dimensional system of point vortices [r. dickman, prl 90, 108701 (2003)]. here i review observational and theoretical aspects of fractal rain distributions and chaotic advection, and present new results on tracer distributions in the vortex model.

Abstract:
Localized rain events have been found to follow power-law distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters et al., PRL 88, 018701 (2002)]. Similar power laws can be generated by treating raindrops as passive tracers advected by the velocity field of a two-dimensional system of point vortices [R. Dickman, PRL 90, 108701 (2003)]. Here I review observational and theoretical aspects of fractal rain distributions and chaotic advection, and present new results on tracer distributions in the vortex model.

Abstract:
We present a result relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. The result is supported by numerical computation of the resonances of the system of n disks on a plane. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.