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Generalized fractal dimensions on the negative axis for compactly supported measures
Author(s) -
Germinet François,
Tcheremchantsev Serguei
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310377
Subject(s) - mathematics , infimum and supremum , measure (data warehouse) , ball (mathematics) , separable space , bounded function , borel measure , fractal , iterated function system , probability measure , pure mathematics , discrete mathematics , mathematical analysis , database , computer science
We study generalized fractal dimensions of measures, called the Hentschel–Procaccia dimensions and the generalized Rényi dimensions. We consider compactly supported Borel measures with finite total mass on a complete separable metric space. More precisely we discuss in great generality finiteness and equality of the different lower and upper dimensions for negative values of their argument q . In particular we do not assume that the measure satisfies to the so called “doubling condition”. A key tool in our analysis is, given a measure μ , the function g ( ε ), ε > 0, defined as the infimum over all points x in the support of μ of the quantity μ ( B ( x , ε )), where B ( x , ε ) is the ball centered at x and of radius ε . We provide counter examples to show the optimality of some criteria for finiteness and equality of the dimensions. We also relate this work to quantum dynamics. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)