Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T12:41:30.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum

Published online by Cambridge University Press:  01 October 2007

JULIEN BARRAL  and
MOUNIR MENSI
JULIEN BARRAL
Affiliation:
INRIA Rocquencourt, 78153 Le Chesnay cedex, France (email: [email protected])
MOUNIR MENSI
Affiliation:
INRIA Rocquencourt, 78153 Le Chesnay cedex, France (email: [email protected]) Département de Mathématiques, Faculté des Sciences de Monastir, Route de l’Environnement, 5000 Monastir, Tunisia (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a class of Gibbs measures on self-affine Sierpiński carpets and perform the multifractal analysis of its elements. These deterministic measures are Gibbs measures associated with bundle random dynamical systems defined on probability spaces whose geometrical structure plays a central role. A special subclass of these measures is the class of multinomial measures on Sierpiński carpets. Our result improves the already known result concerning the multifractal nature of the elements of this subclass by considerably weakening and in some cases even eliminating a strong separation condition of geometrical nature.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1419 - 1443
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barral, J., Coppens, M.-O. and Mandelbrot, B. B.. Multiperiodic multifractal martingale measures. J. Math. Pures Appl. 82 (2003), 1555–1589.CrossRefGoogle Scholar
[2]Bedford, T.. Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Dissertation, University of Warwick, 1984.Google Scholar
[3]Bogenschütz, T. and Gundlach, V. M.. Ruelle’s transfer operator for random subshifts of finite type. Ergod. Th. & Dynam. Sys. 15 (1995), 413–447.CrossRefGoogle Scholar
[4]Brown, G., Michon, G. and Peyrière, J.. On the multifractal analysis of measures. J. Stat. Phys. 66 (1992), 775–790.CrossRefGoogle Scholar
[5]Collet, P., Lebowitz, J. L. and Porzio, A.. The dimension spectrum of some dynamical systems. Proceedings of the symposium on statistical mechanics of phase transitions—mathematical and physical aspects (Trebon, 1986). J. Stat. Phys. 47 (1987), 609–644.CrossRefGoogle Scholar
[6]Falconer, K. J.. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 339–350.CrossRefGoogle Scholar
[7]Falconer, K. J.. The dimension of self-affine fractals II. Math. Proc. Cambridge Philos. Soc. 111 (1992), 169–179.CrossRefGoogle Scholar
[8]Falconer, K. J.. Generalized dimensions of measures on self-affine sets. Nonlinearity 12 (1999), 877–891.CrossRefGoogle Scholar
[9]Fan, A.-H.. Multifractal analysis of infinite products. J. Stat. Phys. 86 (1997), 1313–1336.Google Scholar
[10]Feng, D.-J. and Wang, Y.. A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11 (2005), 107–124.CrossRefGoogle Scholar
[11]Heurteaux, Y.. Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. H. Poincaré Probab. Stastist. 34 (1998), 309–338.CrossRefGoogle Scholar
[12]Kesseböhmer, M.. Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14 (2001), 395–409.CrossRefGoogle Scholar
[13]Khanin, K. and Kifer, Y.. Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. 171 (1996).Google Scholar
[14]Kifer, Y.. Fractals via random iterated function systems and random geometric constructions. Fractal geometry and stochastics (Finsterbergen, 1994). Progr. Probab. 37 (1995), 145–164.Google Scholar
[15]Kifer, Y.. Fractal dimensions and random transformations. Trans. Amer. Math. Soc. 348 (1996), 2003–2038.CrossRefGoogle Scholar
[16]Kifer, Y.. Equilibrium states for random expanding transformations. Random Comput. Dynam. 1 (1992–93), 1–31.Google Scholar
[17]King, J. F.. The singularity spectrum for general Sierpiński carpets. Adv. Math. 116 (1995), 1–8.CrossRefGoogle Scholar
[18]Kingman, J. F. C.. The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 (1968), 499–510.Google Scholar
[19]McMullen, C.. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[20]Ngai, S. M.. A dimension result arising from the L q-spectrum of a measure. Proc. Amer. Math. Soc. 125 (1997), 2943–2951.CrossRefGoogle Scholar
[21]Olsen, L.. A multifractal formalism. Adv. Math. 116 (1995), 82–196.CrossRefGoogle Scholar
[22]Olsen, L.. Self-affine multifractal Sierpiński sponges in . Pacific J. Math. 183 (1998), 143–199.CrossRefGoogle Scholar
[23]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990), 1–268.Google Scholar
[24]Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics). The University of Chicago Press, Chicago, 1997.CrossRefGoogle Scholar
[25]Pesin, Y. and Weiss, H.. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7 (1997), 89–106.CrossRefGoogle ScholarPubMed
[26]Pollicott, M. and Weiss, H.. The dimensions of some self-affine limit sets in the plane and hyperbolic sets. J. Stat. Phys. 77 (1994), 841–866.CrossRefGoogle Scholar
[27]Rand, D. A.. The singularity spectrum f(α) for cookie-cutters. Ergod. Th. & Dynam. Sys. 9 (1989), 527–541.CrossRefGoogle Scholar
[28]Schmeling, J. and Siegmund-Schultze, R.. The singularity spectrum of self-affine fractals with a Bernoulli measure. Preprint, 1992.Google Scholar