Fractal sets

The black region in Fig. 8.2(a) shows the phase-space points in TZ that survive one kick, i.e. a single application of the mapping T. The black regions in Fig. 8.2(b) and (c) represent phase-space points that are not ionized after two and three kicks, respectively. It appears that the black regions in Fig. 8.2 indeed represent the first three stages in the construction of a fractal set i.e. the set of phase-space points that never... 

The generation parameter defining the generation of ionizing trajectories in the self-similar structure in Fig. 10 is related to the number w of encounters of the two electrons at ri = T2 rather than to the ionization time. This interpretation is confirmed in Fig. 11 which shows the density n of trajectories starting with initial conditions uniformly distributed in the middle panel of Fig. 10 as function of the number w of encounters of the two electrons and of the ionization time T. The density n is proportional to minus the derivative of the survival probability with respect to the relevant variable (w or T). The logarithmic plot in Fig. 11a reveals an exponential decay of the density, n(w)ocexp(—0.27w), and hence also of the survival probability, as a function of the number of encounters of the two electrons, just as expected for a self-similar fractal set of trapped trajectories. The doubly logarithmic plot of the density of trajectories in Fig. 11b reveals a power-law decay of the density, (T) oc and hence... 

Use of the concept of the fractal set allows one to examine the dependence of physical properties on the behavior of hierarchical structures. Such structures appear in stochastic inhomogeneous medium. 

Algorithms for the formation of fractal sets and the determination of fractal dimension have a significant place in fractal theory. Simple models of fractal sets are considered below to illustrate formation algorithms and the calculation of the fractal dimension. 

The process of obtaining fractal sets at the transition from order to chaos can be regarded as an example of the change of boundaries between different regions which possess gravity centers (attractors) influencing the distribution of points in the region. Now the boundary constitutes a kind of order-disorder phase transition [20, 21]. 

For such an iteration function scheme, a fractal attractor exists. The best known example for obtaining a fractal set is the square representation in the complex plane... 

The idea that every fractal set needs its own ruler in order to be measured will be used when analyzing the physical properties of heterogeneous media. 

The non empty limited set E C O is called a self-similar set if it may be represented as the union of a limited number of two by two nonoverlapping subsets Ei,i = 1, n (n > 1), such that E is similar to E with coefficient k. An arbitrary segment, the Sierpinski carpet and sponge are examples of self-similar sets. For fractal sets, the Hausdorff-Besicovitch dimension coincides with the self-similar dimension. 

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