Dimension of a measure

• Autorzy: Pertti Mattila , Manuel Morán , José-Manuel Rey
• Języki publikacji: EN

Abstrakty

EN

We propose a framework to define dimensions of Borel measures in a metric space by formulating a set of natural properties for a measure-dimension mapping, namely monotonicity, bi-Lipschitz invariance, (σ-)stability, etc. We study the behaviour of most popular definitions of measure dimensions in regard to our list, with special attention to the standard correlation dimensions and their modified versions.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 142
• Numer: 3
• Strony: 219-233

Daty

wydano

2000

otrzymano

1999-02-04

poprawiono

1999-04-29

Twórcy

autor

Pertti Mattila

• Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland

autor

Manuel Morán

• Departamento de Análisis Económico, Universidad Complutense, Campus de Somosaguas, 28223 Madrid, Spain

autor

José-Manuel Rey

• Departamento de Análisis Económico, Universidad Complutense, Campus de Somosaguas, 28223 Madrid, Spain

Bibliografia

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• [3] K. J. Falconer, Fractal Geometry--Mathematical Foundations and Applications, Wiley, 1990.
• [4] K. J. Falconer, Techniques in Fractal Geometry, Wiley, 1998.
• [5] P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50 (1983), 346-349.
• [6] M. de Guzmán, Differentiation of Integrals in $ℝ^n$, Lecture Notes in Math. 481, Springer, 1975.
• [7] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995.
• [8] Ya. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Statist. Phys. 71 (1993), 529-547.
• [9] L.-S. Young, Dimension, entropy and Liapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124.