Dimension of a measure

Pertti Mattila; Manuel Morán; José-Manuel Rey

Studia Mathematica (2000)

  • Volume: 142, Issue: 3, page 219-233
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Mattila, Pertti, Morán, Manuel, and Rey, José-Manuel. "Dimension of a measure." Studia Mathematica 142.3 (2000): 219-233. <http://eudml.org/doc/216799>.

@article{Mattila2000,
abstract = {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.},
author = {Mattila, Pertti, Morán, Manuel, Rey, José-Manuel},
journal = {Studia Mathematica},
keywords = {Borel measures; monotonicity; bi-Lipschitz invariance; stability; measure dimensions; correlation dimensions},
language = {eng},
number = {3},
pages = {219-233},
title = {Dimension of a measure},
url = {http://eudml.org/doc/216799},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Mattila, Pertti
AU - Morán, Manuel
AU - Rey, José-Manuel
TI - Dimension of a measure
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 3
SP - 219
EP - 233
AB - 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.
LA - eng
KW - Borel measures; monotonicity; bi-Lipschitz invariance; stability; measure dimensions; correlation dimensions
UR - http://eudml.org/doc/216799
ER -

References

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  1. [1] L. Barreira, Ya. Pesin and J. Schmeling, Dimensions of hyperbolic measures--a proof of the Eckmann-Ruelle conjecture, Ann. of Math., to appear. Zbl0871.58054
  2. [2] C. D. Cutler, Some results on the behaviour and estimation of the fractal dimensions of distributions on attractors, J. Stat. Phys. 62 (1990), 651-708. Zbl0738.58029
  3. [3] K. J. Falconer, Fractal Geometry--Mathematical Foundations and Applications, Wiley, 1990. Zbl0689.28003
  4. [4] K. J. Falconer, Techniques in Fractal Geometry, Wiley, 1998. Zbl0869.28003
  5. [5] P. Grassberger and I. Procaccia, Characterization of strange attractors, Phys. Rev. Lett. 50 (1983), 346-349. 
  6. [6] M. de Guzmán, Differentiation of Integrals in ℝ n , Lecture Notes in Math. 481, Springer, 1975. 
  7. [7] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995. Zbl0819.28004
  8. [8] Ya. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Statist. Phys. 71 (1993), 529-547. Zbl0916.28006
  9. [9] L.-S. Young, Dimension, entropy and Liapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), 109-124. 

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