Real Analysis Exchange

Typical properties of correlation dimension.

Józef Myjak and Ryszard Rudnicki

Full-text: Open access

Abstract

Let \((X,\rho)\) be a complete separable metric space and \(\mathcal M\) be the set of all probability Borel measures on \(X\). We show that if the space \(\mathcal M\) is equipped with the weak topology, the set of measures having the upper (resp. lower) correlation dimension zero is re\-si\-dual. Moreover, the upper correlation dimension of a typical (in the sense of Baire category) measure is estimated by means of the local lower entropy and local upper entropy dimensions of \(X\).

Article information

Source
Real Anal. Exchange, Volume 28, Number 2 (2002), 269-278.

Dates
First available in Project Euclid: 20 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184963795

Mathematical Reviews number (MathSciNet)
MR2009754

Zentralblatt MATH identifier
1048.37020

Subjects
Primary: 28A80: Fractals [See also 37Fxx]
Secondary: 54E52: Baire category, Baire spaces

Keywords
measure dimension residual subset

Citation

Myjak, Józef; Rudnicki, Ryszard. Typical properties of correlation dimension. Real Anal. Exchange 28 (2002), no. 2, 269--278. https://projecteuclid.org/euclid.rae/1184963795


Export citation

References

  • W. Chin, B. Hunt and J. A. Yorke, Correlation dimension for iterated function systems, Trans. Amer. Math. Soc., 349 (1997), 1783–1796.
  • P. M. Gruber, Dimension and structure of typical compact sets, continua and curves, Mh. Math., 108 (1989), 149–164.
  • C. A. Guerin and M. Holschneider, On equivalent definitions of the correlation dimension for a probability measure, J. Statist. Phys., 86 (1997), 707–720.
  • A. Manning and K. Simon, A short existence proof for correlation dimension, J. Statist. Phys., 90 (1998), 1047–1049.
  • Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Stat. Phys, 71 (1993), 529–547.
  • Ya. B. Pesin and A. Tempelman, Correlation dimension of measures invariant under group actions, Random Comput. Dynam., 3 (1995), 137–156.
  • Ya. B. Pesin and H. Weiss, The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7 (1997), 89–106.
  • Ya. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lecture in Math. Series, Chicago, 1997.
  • I. Procaccia, P. Grassberger and H.G.E. Hentschel, On the characterization of chaotic motions, Dynamical systems and chaos (Sitges/Barcelona, 1982),
  • T. D. Sauer and J. A. Yorke, Are the dimensions of a set and its image equal under typical smooth functions, Ergodic Theory Dynam. Systems, 17 (1997), 941–956.
  • K. Simon and B. Solomyak, Correlation dimension for self-similar Cantor sets with overlaps, Fund. Math., 155 (1998), 293–300.