Abstract
We construct a family of measures called infinite products which generalize Gibbs measures in the one-dimensional lattice gas model. The multifractal properties of these measures are studied under some regularity conditions. In particular, if the τ-function is differentiable, we prove a formula which gives the Hausdorff dimension and packing dimension of the set of singularity points of a given order. Mathematical examples include Riesz products,g-measures, andG-measures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Bohr and D. Rand, The entropy function for characteristic exponents,Physica 25D:387–398 (1987).
R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer-Verlag, Berlin, 1975).
G. Brown and A. H. Dooley, Odometer actions onG-measures,Ergod. Theory Dynam. Syst. 11:279–307 (1991).
G. Brown, G. Michon, and J. Peyrière, On the multifractal analysis of measures,J. Stat. Phys. 66:775–790 (1992).
R. Cawley and R. D. Mauldin, Multifractal decompositions of Moran fractals,Adv. Math. 92:196–236 (1992).
P. Collet, Hausdorff dimension of singularities for invariant measures of expanding dynamical systems, inDynamical Systems Valparaiso 1986, R. Bamón et al., eds. (Springer-Verlag, Berlin, 1988).
P. Collet, J. L. Lebowitz, and A. Porzio, The dimension spectrum of some dynamical systems,J. Stat. Phys. 47:609–644 (1987).
R. S. Ellis, Large deviations for a class of random vectors,Ann. Prob. 12:1–12 (1984).
A. H. Fan, Sur les dimensions de mesures,Studia Math. 111(1):1–17 (1994).
A. H. Fan, On uniqueness ofG-measures andg-measures,Studia Math. 119(3):255–269 (1996).
A. H. Fan, Ergodicity, unidimensionality and multifractality of self-similar measures,Kyushu J. Math., to appear.
A. H. Fan, Analyse multifractale de certains produits de Riesz,C. R. Acad. Sci. Paris I 321:399–404 (1995).
U. Frisch and G. Parisi, On the singularity structure of fully developed turbulence, appendix to U. Frisch, Fully developed turbulence and intermittence, inTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (North-Holland, Amsterdam, 1985), pp. 84–88.
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. shraiman, Fractal measures and their singularities: The characterisation of strange sets,Phys. Rev. A 33:1141 (1986).
M. Keane, Strongly mixingg-measures,Inv. Math. 16:309–324 (1974).
J. K. C. Kingman,Subadditive Processes (Springer, Berlin 1976).
K. S. Lau, Fractal measures and the meanp-variations convolutions,J. Funct. Anal. 108:427–457 (1992).
K. S. Lau and S. M. Ngai, Multifractal measure and a weak separation condition, preprint.
K. S. Lau and J. R. Wang, Mean quadratic variations and Fourier asymptotics of selfsimilar measures,Monatsh. Math. 115:99–132 (1993).
P. Mattila,Geometry of Sets and Measures in Euclidean Space (Cambridge University Press, Cambridge, 1995).
B. Mandelbrot, Possible refinement of the log-normal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, inStatistical Models and Turbulence (Springer-Verlag, Berlin, 1972), pp. 333–351.
L. Olsen, A multifractal formalism,Adv. Math., to appear.
K. Petersen,Ergodic Theory (Cambridge University Press, Cambridge, 1983).
J. Peyrière, Études de quelques propriétés des produits de Riesz,Ann. Inst. Fourier 25:127–169 (1975).
D. Rand, The singularity spectrumf(α) for cookie-cutters,Ergod. Theory Dynam. Syst. 9:527–541 (1989).
R. T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, New Jersey, 1970).
C. R. T. Roger,Hausdorff Measures (Cambridge University Press, Cambridge, 1970).
D. Ruelle,Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics (Addison-Wesley, Readings, Massachusetts, 1978).
D. Simplaere, Thèse, Université de Paris VI (1992).
R. S. Strichartz, Self-similar measures and their Fourier transforms I,Indiana Univ. Math J. 39:155–186 (1989).
R. S. Strichartz, Self-similar measures and their Fourier transforms III,Indiana Univ. Math J. 42:367–411 (1989).
M. Tamashiro, Dimension in a separable metric space, preprint.
T. Tel, Fractals and multifractals,Z. Naturforsch. 43A:1154–1174 (1988).
C. Tricot, Two definitions of fractional dimension,Math. Proc. Camb. Philos. Soc. 91:57–74 (1982).
A. Zygmund,Trigonometric Series, Vols. 1 & 2, 2nd ed. (Cambridge University Press, Cambridge, 1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hua, F.A. Multifractal analysis of infinite products. J Stat Phys 86, 1313–1336 (1997). https://doi.org/10.1007/BF02183625
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02183625