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Fractals, Multifractals, and Thermodynamics

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Abstract

The basic concept of fractals and multifractals are introduced for pedagogical purposes, and the present status is reviewed. The emphasis is put on illustrative examples with simple mathematical structures rather than on numerical methods or experimental techniques. As a general characteriza­tion of fractals and multifractals a thermodynamical formalism is introduced, establishing a connec­tion between fractal properties and the statistical mechanics of spin chains.
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... where D q refers to the order q generalized dimension (Tel 1988), I q e ð Þ is called the generalized entropy (R enyi 1961;Appleby 1996), and P i e ð Þ q is the spatial probability of the ith fractal units in the linear scale of e: In general, this parameter can characterize the scale invariance of the generalized entropy, and the order q plays a role in adjusting the contribution of each fractal unit to the overall spatial probability. ...
... For a strictly fractal system, the singularity exponents of its subsystems should remain constant across all scales. For random fractals in urban systems, however, these are generated by nondeterministic rules and accordingly show more randomness and anomalies (Tel 1988), with the singularity exponents changing with the scales. Consequently, they become unlikely to estimate. ...
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... Additionally, significant differences in the long-term crosscorrelation were accompanied by changes in the degree of multifractality, in most cases. A possible explanation could be that multifractality results from more complex dynamics (Tel, 1988) which tend to vary more from region to region. On the other hand, this contradicts the findings of our previous resting-state study, where H(2) values varied the most [cf. ...
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