Abstract
The short-distance behavior of the measure of a sphere and of the correlation integral is determined, in the case of disconnected repellers, by scaling laws whose corrections are oscillating functions, periodic or aperiodic, depending on exact or approximate self-similarity of the measure. The Mellin transforms prove to be the correct analytic tool in order to investigate these corrections to scaling. It has been previously proved that they are meromorphic for linear Cantor sets and that the leading pole gives the correlation dimension in agreement with the results of the thermodynamic formalism. Here we show that the residues of these poles can also be computed to any desired accuracy with simple algorithms and that the knowledge of the singularity spectrum of the Mellin transforms provides the Fourier spectrum of the scaling correction for the self-similar measure and that it reproduces the damped oscillations in the generic case. The method applies to the nonlinear repellers such as the disconnected Julia sets by using an approximation theorem.
Similar content being viewed by others
References
E. N. Lorenz, Deterministic nonperiodic flow,J. Atmos. Sci. 20:130–141 (1963).
M. Hénon, A two dimensional mapping with a strange attractor,Commun. Math. Phys. 50:69–77 (1976).
M. Widom, D. Bensimon, L. P. Kadanoff, and S. J. Shenker, Strange objects in the complex plane,J. Stat. Phys. 32:443 (1983).
D. Ruelle, Measures describing a turbulent flow,N. Y. Acad. Sci. 357:1–9 (1980).
D. Ruelle,Chaotic Evolution and Strange Attractors (Cambridge University Press, Cambridge, 1989).
D. Ruelle, Repellers for real analytic maps,Ergodic Theory Dynam. Syst. 2:99–107 (1982).
H. Brolin, Invariant sets under iteration of rational functions,Arch. Math. 6:103–144 (1965).
D. Ruelle, Thermodynamical formalism,Encyclopedia of Mathematics and its Applications, Vol. 5 (Addison-Wesley, Reading, Massachusetts, 1978).
P. Collet, J. L. Lebowitz, and A. Porzio, The dimension spectrum of some dynamical systems,J. Stat. Phys. 47:609–644 (1987).
E. Vul, Y. Sinai, and K. Khanin, Feigenbaum universality and the thermodynamic formalism,Russ. Math. Surv. 39(3):1–40 (1984).
S. Vaienti, Generalized spectra for the dimensions of strange sets,J. Phys. A 21:2313–2320 (1988).
G. Servizi, G. Turchetti, and S. Vaienti, Generalized dynamical variables and measures for the Julia sets,Nuovo Cimento B 101:285 (1988).
G. Turchetti and S. Vaienti, Analytical estimates of fractal and dynamical properties of one dimensional expanding maps,Phys. Lett. A 128:343–348 (1988).
S. Vaienti, Some properties of mixing repellers,J. Phys. A 21:2023–2043 (1988).
R. Badii and A. Politi, Intrinsic oscillations in measuring the fractal dimension,Phys. Lett. A 104:303–305 (1984).
L. A. Smith, J. D. Fournier, and L. A. Spiegel, Lacunarity and intermittency in fluid turbulence,Phys. Lett. A 114:465–468 (1986).
D. Bessis, G. Servizi, G. Turchetti, and S. Vaienti, Mellin transforms and correlation dimensions,Phys. Lett. A 119:345–347 (1987).
D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, and S. Vaienti, Mellin transforms of correlation integrals and generalized dimensions of strange sets,Phys. Rev. A 36:920–928 (1987).
D. Bessis, J. S. Jeronimo, and P. Moussa, Mellin transform associated with Julia sets and physical applications,J. Stat. Phys. 34:75–110 (1984).
J. D. Fournier, G. Turchetti, and S. Vaienti, Singularity spectrum of the generalized energy integrals,Phys. Lett. A 140:331 (1989).
E. Orlandini, G. Servizi, M. C. Tesi, and G. Turchetti, Singularities of the energy integrals and scaling laws of the dimension spectra,Nuovo Cimento, to appear.
L. S. Young, Dimension, entropy and Lyapunov exponents,Ergodic Theory Dynam. Syst. 2:109–124 (1982).
T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of real and integers, University of Delft, preprint (1990).
T. C. Hasley, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. Shraiman, Fractal measures and their singularities: The characterization of strange sets,Phys. Rev. A 33:1141–1151 (1986).
M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals,Proc. R. Soc. Lond. A 399:243–275 (1985).
J. H. Elton, Ergodic theorem for iterated maps,Ergodic Theory Dynam. Syst. 7:481–488 (1987).
K. J. Falconer,The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Orlandini, E., Tesi, M.C. & Turchetti, G. Meromorphic structure of the Mellin transforms and short-distance behavior of correlation integrals. J Stat Phys 66, 515–533 (1992). https://doi.org/10.1007/BF01060078
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01060078