Phase-space fractal and powerlaw decay

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 Hausdorff-Besicovitch dimension is now defined as the critical value do at which jumps from 0 to oo, i.e. 

The definition (8.2.2) of the Hausdorff-Besicovitch dimension can be applied to any set, fractal or nonfractal, scaling or nonscaling. Applied to our fractal A however, we obtain a paradox. It turns out that the 

Hausdorff-Besicovitch dimension of A is 2, although its area in phase space converges to zero as the number of kicks approaches infinity. In order to resolve this paradox, the definition (8.2.2) has to be extended by including logarithmic corrections (Hausdorff (1919), Umberger et al. (1986)). The idea is to retain the general structure of (8.2.1), but to admit a larger class of functions than to counterbalance the proliferation of the number of boxes B e) for e 0. We define 

The function G e) is called the gauge function . In the sense of a test function, the gauge function is in principle arbitrary, but not all choices of G yield a finite measure //. In order to find a suitable gauge function the following expansion turned out to be useful  

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