Mellin transforms associated with Julia sets and physical applications. (English) Zbl 0602.58028

We introduce the Mellin transform of the balanced invariant measure associated to the Julia set generated by a rational transformation. We show that its analytic continuation is a meromorphic function, the poles of which are on a semi-infinite periodic lattice. This allows one to have an understanding of the behavior of the measure near a repulsive fixed point. Trace identities corresponding to the fact that the analytically continued Mellin transform vanishes at negative integers are derived for the polynomial case.
The quadratic map is first analyzed in detail, and the analytic properties of the inverse of the Green’s function are exhibited. Of interest is the appearance of a dense set of spikes at dyadic points when the Julia set is disconnected. These results are used to study the residues of the Mellin transform. A certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems, the excitation spectrum of which displays unusual oscillations. The appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion.


37A99 Ergodic theory
28D05 Measure-preserving transformations
28A75 Length, area, volume, other geometric measure theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37N99 Applications of dynamical systems
81Q99 General mathematical topics and methods in quantum theory
Full Text: DOI


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