Proceedings of the American Mathematical Society

Author(s): Jouni Luukkainen; Eero Saksman
Journal: Proc. Amer. Math. Soc. 126 (1998), 531-534.
MSC (1991): Primary 28A12; Secondary 54F45
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Abstract: We prove that a complete metric space $X$

carries a doubling measure if and only if $X$

is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on $X$

and of the homogeneity exponents of $X$

are equal. We also show that a closed subset $X$

of $\mathbf{R}^{n}$ carries a measure of homogeneity exponent $n$

. These results are based on the case of compact $X$

due to Vol $^{\prime }$ berg and Konyagin.

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Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28A12, 54F45

Jouni Luukkainen
Affiliation: Department of Mathematics, P.O. Box~4 (Yliopistonkatu~5), FIN-00014 University of Helsinki, Finland
Email: luukkain@cc.helsinki.fi

Eero Saksman
Affiliation: Department of Mathematics, P.O. Box~4 (Yliopistonkatu~5), FIN-00014 University of Helsinki, Finland
Email: saksman@cc.helsinki.fi

DOI: 10.1090/S0002-9939-98-04201-4
PII: S 0002-9939(98)04201-4
Keywords: Doubling metric space, homogeneous metric space, Assouad dimension, doubling measure, homogeneous measure
Received by editor(s): August 20, 1996
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1998, American Mathematical Society

David A. Herron and Volker Mayer, Bi-Lipschitz group actions and homogeneous Jordan curves, Illinois J. Math. 43 (1999), 770--792. MR CMP 1 712 522

Eero Saksman, Remarks on the nonexistence of doubling measures, Ann. Acad. Sci. Fenn. Math. 24 (1999), 155-163. MR 2000b:28006

Guy David and Stephen Semmes, Fractured fractals and broken dreams. Self-similar geometry through metric and measure, Oxford Lecture Ser. Math. Appl., vol. 7, Clarendon Press, Oxford Univ. Press, New York, Oxford, 1997. MR 99h:28018

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