Skip to main content
SpringerLink
Log in

On the Hausdorff Dimension of Invariant Measures of Weakly Contracting on Average Measurable IFS

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dudley, R.M.: Probabilities and Metrics. Aarhus Universitet, Aarhus (1976)

    MATH  Google Scholar 

  2. Elton, J.H.: An ergodic theorem for iterated maps. Ergod. Theory Dyn. Syst. 7, 481–488 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fan, A.H., Simon, K., Toth, H.R.: Contracting on average random IFS with repelling fixpoint. J. Stat. Phys. 122, 169–193 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Fortet, R., Mourier, B.: Convergence de la répartition empirique vers la répartition théorétique. Ann. Sci. Éc. Norm. Super. 70, 267–285 (1953)

    MATH  MathSciNet  Google Scholar 

  5. Johansson, A., Öberg, A.: Exact dimension of Cantor type measures generated by iterated function systems. Preprint

  6. Myjak, J., Szarek, T.: On Hausdorff dimension of invariant measures arising from non-contractive iterated function systems. Ann. Mat. Pura Appl. 181, 223–237 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Nicol, M., Sidorov, N., Broomhead, D.: On the fine structure of stationary measures in systems which contract-on-average. J. Theor. Probab. 15, 715–730 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Rams, M.: Dimension estimates for invariant measures of contract ing-on-average iterated function systems. Preprint

  9. Szarek, T.: The dimension of self-similar measures. Bull. Pol. Acad. Sci. Math. 48, 293–302 (2000)

    MATH  MathSciNet  Google Scholar 

  10. Werner, I.: Contractive Markov systems. J. Lond. Math. Soc. 71(2), 236–258 (2005)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland

    Joanna Jaroszewska

  2. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950, Warszawa, Poland

    Michał Rams

Authors
  1. Joanna Jaroszewska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michał Rams
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michał Rams.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jaroszewska, J., Rams, M. On the Hausdorff Dimension of Invariant Measures of Weakly Contracting on Average Measurable IFS. J Stat Phys 132, 907–919 (2008). https://doi.org/10.1007/s10955-008-9566-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9566-3

Keywords

Search

Navigation