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Computability of the packing measure of totally disconnected self-similar sets

Published online by Cambridge University Press:  06 February 2015

MARTA LLORENTE  and
MANUEL MORÁN
MARTA LLORENTE
Affiliation:
Dpt. Análisis Económico: Economía cuantitativa, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain email m.llorente@uam.es
MANUEL MORÁN
Affiliation:
Dpt. Análisis Económico I, Universidad Complutense de Madrid, Campus de Somosaguas, 28223 Madrid, Spain email mmoranca@ccee.ucm.es
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Abstract

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016 , pp. 1534 - 1556
Copyright
© Cambridge University Press, 2015 

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References

Ayer, E. and Strichartz, R.. Exact Hausdorff measure and intervals of maximal density for Cantor sets. Trans. Amer. Math. Soc. 351(9) (1999), 3725–3741.CrossRefGoogle Scholar
Baek, H. K.. Packing dimension and measure of homogeneous Cantor sets. Bull. Aust. Math. Soc. 74(3) (2006), 443–448.Google Scholar
Baoguo, J. and Zhiwei, Z.. The packing measure of a class of generalized sierpinski carpet. Anal. Theory Appl. 20(1) (2004), 69–76.CrossRefGoogle Scholar
Baoguo, J., Zuoling, Z., Zhiwei, Z. and Jun, L.. The packing measure of the Cartesian product of the middle third Cantor set with itself. J. Math. Anal. Appl. 288 (2003), 424–441.Google Scholar
Feng, D.. Exact packing measure of linear Cantor sets. Math. Nachr. 248/249 (2003), 102–109.Google Scholar
Feng, D. and Hua, J. G.. Some relations between packing premeasure and packing measure. Bull. Lond. Math. Soc. 31 (1998), 665–670.Google Scholar
Garcia, I. and Zuberman, L.. Exact packing measure of central Cantor sets in the line. J. Math. Anal. Appl. 386(2) (2012), 801–812.Google Scholar
Llorente, M. and Morán, M.. Self-similar sets with optimal coverings and packings. J. Math. Anal. Appl. 334 (2007), 1088–1095.Google Scholar
Llorente, M. and Moran, M.. Advantages of the centered Hausdorff measure from the computability point of view. Math. Scand. 107(1) (2010), 103–122.CrossRefGoogle Scholar
Llorente, M. and Moran, M.. An algorithm for computing the centered Hausdorff measures of self-similar sets. Chaos Solitons Fractals 45(3) (2012), 246–255.Google Scholar
Mattila, P.. On the structure of self-similar fractals. Ann. Acad. Sci. Fenn. Ser. A I Math. 7(2) (1982), 189–195.Google Scholar
Morán, M.. Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle. Nonlinearity 18(2) (2005), 559–570.Google Scholar
Sullivan, D.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.CrossRefGoogle Scholar
Taylor, S. J. and Tricot, C.. Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.Google Scholar
Taylor, S. J. and Tricot, C.. Packing measure of rectifiable subsets of the plane. Math. Proc. Cambridge Philos. Soc. 99 (1986), 285–296.Google Scholar
Tricot, C.. Sur la classification des ensembles boréliens de mesure de Lebesgue nulle. These de doctorat, Geneve, 1979.Google Scholar
Tricot, C.. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar
Tricot, C.. Geometries et Mesures Fractales: Une Introduction. Ellipses, France, 2008, p. 339.Google Scholar
Zhou, Z.. A new estimate of the Hausdorff measure of the Sierpinski gasket. Nonlinearity 13 (2000), 479–491.Google Scholar