Abstract
We study the doubling property of binomial measures on the middle interval Cantor set. We obtain a necessary and sufficient condition that enables a binomial measure to be doubling. Then we determine those doubling binomial measures which can be extended to be doubling on [0,1]. Finally, we construct a compact set X in [0,1] and a doubling measure μ on X, such that \(\overline{F}_{X}=X\) and \({\mu|}_{E_{X}}\) is doubling on E X , where E X is the set of accumulation points of X and F X is the set of isolated points of X.
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References
D. Han, L. S. Wang and S. Y. Wen, Thickness and thinness of uniform Cantor sets for doubling measures, Nonliearity, 22 (2009), 545–551.
R. Kaufman and J. M. Wu, Two problems on doubling measures, Rev. Math. Iberoamericana, 11 (1995), 527–545.
M. L. Lou and M. Wu, Doubling measure with doubling continuous part, Proc. Amer. Math. Soc., 138 (2010), 3585–3589.
J. Luukkainen and E. Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc., 126 (1998), 531–534.
L. Olsen, A multifractal formalism, Adv. Math., 116 (1995), 82–196.
E. Saksman, Remarks on the nonexistence of doubling measures, Ann. Acad. Sci. Fenn. Math., 24 (1999), 155–163.
S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about A ∞-weights, Rev. Math. Iberoamericana, 12 (1996), 337–410.
A. L. Vol’berg and S. V. Konyagin, On measures with the doubling condition, Math. USSR-Izv., 30 (1988), 629–638.
X. H. Wang, S. Y. Wen and Z. X. Wen, Doubling measures on Whitney modification sets, J. Math. Anal. Appl., 342 (2008), 761–765.
X. H. Wang, S. Y. Wen and Z. X. Wen, Doubling measures and non-quasisymmetric maps on Whitney modification sets in Euclidean spaces, Illinois J. Math., 52 (2008), 1291–1300.
J. M. Wu, Null sets for doubling and dyadic doubling measures, Ann. Acad. Sci. Fenn. Math., 18 (1993), 77–91.
P. L. Yung, Doubling properties of self-similar measures, Indiana Univ. Math. J., 56 (2007), 965–990.
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Wang, X., Wen, S. Doubling measures on Cantor sets and their extensions. Acta Math Hung 134, 431–438 (2012). https://doi.org/10.1007/s10474-011-0186-z
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DOI: https://doi.org/10.1007/s10474-011-0186-z