On Multifractals: Thermodynamics and Critical Exponents
Using the two-scale Cantor set as a starting point the thermodynamical formalism is discussed, including entropy, free energy, and dimensions. The connection to the f — α spectrum is illustrated, in particular when the probability measure is a Gibb’s ensemble. Regarding transitions to chaos as phase transitions critical exponents is defined. The way these enter the thermodynamics is outlined. As a main result the general interpretation of the cross-over scale as the scale where the system change from a fractal to a nonfractal behavior is shown to be ambiguous. As a representative example the global quasi-periodic transition (including noise) described by the sine map is treated.
KeywordsCritical Exponent Hausdorff Dimension Entropy Function Characteristic Exponent Natural Measure
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