Crossover scaling for moments in multifractal systems.

Abstract

Invoking the formalism known from second-order phase transitions and thermodynamics, we analyze the step structure obtained at transitions to chaos in dynamical systems or where Cantor sets evolve in general. As examples, we treat the skew tent map analytically and Arnold's sine map numerically, but the presented formalism employed for embedding dimension d=1 is readily extended to higher dimensions. We outline the scaling behavior for the counting, the measure, and higher moments. In particular, we consider the crossover exponent \ensuremath{\nu} which enters the scaling functions and for the measure is related to the critical exponent \ensuremath{\beta} and fractal dimension D. We emphasize that the general presence of a multifractal structure results in a value of \ensuremath{\nu} which depends on from which moment it is defined, and deduce the saturation value of \ensuremath{\nu} in the high-moment limit. Also, we derive the connection to thermodynamical functions as pressure, entropy, and escape rate. Finally, we examine the scaling behavior of the moments and scaling relations for exponents when either a ``ghost'' field or noise is introduced as a conjugated field involving the critical exponents \ensuremath{\alpha}, \ensuremath{\gamma}, and \ensuremath{\delta} as well as the crossover exponent \ensuremath{\mu}.