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The singularity spectrum of the inverse of cookie-cutters

Published online by Cambridge University Press:  01 August 2009

JULIEN BARRAL  and
STÉPHANE SEURET
JULIEN BARRAL
Affiliation:
INRIA Rocquencourt, 78153 Le Chesnay Cedex, France (email: [email protected])
STÉPHANE SEURET
Affiliation:
Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est, CNRS UMR 8050-61, avenue du Général de Gaulle, 94010 Créteil Cedex, France (email: [email protected])
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Abstract

Gibbs measures μ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. We compute the singularity spectrum of the inverse measure of μ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism and its Lq-spectrum possesses one point of non-differentiability. The results rely on heterogeneous ubiquity theorems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1075 - 1095
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Aversa, V. and Bandt, C.. The multifractal spectrum of discrete measures. 18th Winter School on Abstract Analysis (Srn, 1990). Acta Univ. Carolin. Math. Phys. 31 (1990), 5–8.Google Scholar
[2]Barral, J. and Seuret, S.. Sums of Dirac masses and conditioned ubiquity. C. R. Math. Acad. Sci. Paris 339(11) (2004), 787–792.CrossRefGoogle Scholar
[3]Barral, J. and Seuret, S.. Combining multifractal additive and multiplicative chaos. Comm. Math. Phys. 257(2) (2005), 473–497.Google Scholar
[4]Barral, J. and Seuret, S.. Heterogeneous ubiquitous systems in R d and Hausdorff dimensions. Bull. Braz. Math. Soc. 38(3) (2007), 467–515.CrossRefGoogle Scholar
[5]Barral, J. and Seuret, S.. The multifractal nature of heterogeneous sums of Dirac masses. Math. Proc. Cambridge Philos. Soc. 144(3) (2008), 707–727.Google Scholar
[6]Barral, J. and Seuret, S.. Ubiquity and large intersections properties under digit frequencies constraints. Math. Proc. Cambridge Philos. Soc. 145 (2008), 527–548.CrossRefGoogle Scholar
[7]Barreira, L., Saussol, B. and Schmeling, J.. Higher-dimensional multifractal analysis. J. Math. Pures Appl. 81 (2002), 67–91.Google Scholar
[8]Ben Nasr, F.. Analyse multifractale de mesures. C. R. Acad. Sci. Paris Série I. Math. 319 (1994), 807–810.Google Scholar
[9]Bowen, R.. Equilibrium States and the Ergodic Theory of Anasov Diffeomorphisms (Springer Lecture Notes in Mathematics, 470). Springer, New York, 1975.Google Scholar
[10]Brown, G., Michon, G. and Peyrière, J.. On the multifractal analysis of measures. J. Stat. Phys. 66 (1992), 775–790.Google Scholar
[11]Dodson, M. M., Melián, M. V., Pestana, D. and Vélani, S. L.. Patterson measure and Ubiquity. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), 37–60.Google Scholar
[12]Durand, A.. Ubiquitous systems and metric number theory. Adv. Math. 218(2) (2008), 368–394.Google Scholar
[13]Falconer, K. J.. Techniques in Fractal Geometry. Wiley, New York, 1997.Google Scholar
[14]Falconer, K. J.. Representation of families of sets by measures, dimension spectra and Diophantine approximation. Math. Proc. Cambridge Philos. Soc. 128 (2000), 111–121.Google Scholar
[15]Fan, A.-H. and Feng, D.-J.. On the distribution of long-term average on the symbolic space. J. Stat. Phys. 99 (2000), 813–856.CrossRefGoogle Scholar
[16]Feng, D.-J., Lau, K. S. and Wu, J.. Ergodic limits on the conformal repellers. Adv. Math. 169(1) (2002), 58–91.Google Scholar
[17]Jaffard, S.. Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3 (1997), 1–22.Google Scholar
[18]Jaffard, S.. On lacunary wavelet series. Ann. Appl. Probab. 10(1) (2000), 313–329.CrossRefGoogle Scholar
[19]Mandelbrot, B. and Riedi, R.. Exceptions to the multifractal formalism for discontinuous measures. Math. Proc. Cambridge Philos. Soc. 123 (1998), 133–157.Google Scholar
[20]Olsen, L.. A multifractal formalism. Adv. Math. 116 (1995), 82–196.CrossRefGoogle Scholar
[21]Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics). The University of Chicago Press, Chicago, IL, 1997.CrossRefGoogle Scholar
[22]Pesin, Y. and Weiss, H.. The multifractal analysis of Gibbs measures: motivation, mathematical moundation, and examples. Chaos 7 (1997), 89–106.CrossRefGoogle ScholarPubMed
[23]Peyrière, J.. A vectorial multifractal formalism. Fractal Geometry and Applications (Proceedings of Symposia in Pure Mathematics, 72(2)). Eds. M. L. Lapidus and M. van Frankenhuijsen. American Mathematical Society, Providence, RI, 2004, pp. 217–230.Google Scholar
[24]Rand, D. A.. The singularity spectrum f(α) for cookie-cutters. Ergod. Th. & Dynam. Sys. 9 (1989), 527–541.CrossRefGoogle Scholar