Skip to main content
SpringerLink
Log in

Algorithms for approximation of invariant measures for IFS

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

We prove convergence of two algorithms approximating invariant measures to iterated function systems numerically. We consider IFSs with finitely many continuous and injective non-overlapping maps on the unit interval. The first algorithm is a version of the Ulam algorithm for IFSs introduced by Strichartz et al. [16]. We obtain convergence in the supremum metric for distribution functions of the approximating eigen-measures to a unique invariant measure for the IFS. We have to make some modifications of the usual way of treating the Ulam algorithm due to a problem concerning approximate eigenvalues, which is part of our more general situation with weights not necessarily being related to the maps of the IFS. The second algorithm is a new recursive algorithm which is an analogue of forward step algorithms in the approximation theory of ODEs. It produces a sequence of approximating measures that converges to a unique invariant measure with geometric rate in the supremum metric. The main advantage of the recursive algorithm is that it runs much faster on a computer (using Maple) than the Ulam algorithm.

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. Baladi, V., Holschneider, M.: Approximation of nonessential spectrum of tranfer operators. Nonlinearity 12, 525–538 (1999)

    Google Scholar 

  2. Baladi, V., Isola, S., Schmitt, B.: Transfer operators for piecewise affine approximations of interval maps. Ann. Inst. Henri Poincaré (phys. théor.) 62, 553–574 (1995)

    Google Scholar 

  3. Berger, N., Hoffman, C., Sidoravicius, V.: Nonuniqueness for specifications in ℓ2+∈. Preprint available on http://www.arxiv.org (PR/0312344 v2, 18 Dec 2003)

  4. Bramson, M., Kalikow, S.: Nonuniqueness in g-functions. Israel J. Math. 84, 153–160 (1993)

    MATH  Google Scholar 

  5. Froyland, G., Aihara, K.: Rigorous numerical estimation of Lyapunov exponents and invariant measures of iterated function systems and random matrix products. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (1), 103–122 (2000)

    Article  Google Scholar 

  6. Góra, P., Boyarsky, A.: Compactness of invariant densities for families of expanding, piecewise monotonic transformations. Can. J. Math. 41, 855–869 (1989)

    Google Scholar 

  7. Góra, P., Boyarsky, A.: Laws of Chaos Boston: Birkhäuser, 1997

  8. Hennion, H.: Sur un théorème spectral et son application aux noyaux lipschitziens. Proc. Am. Math. Soc. 118, 627–634 (1993)

    MATH  Google Scholar 

  9. Hervé, L.: Étude d’operateurs quasi compacts positifs. Applications aux opérateurs de transfert. Ann. Inst. Henri Poinaré 30, 437–466 (1994)

    Google Scholar 

  10. Ionescu-Tulcea, C.T., Marinescu, G.: Théorie ergodique pour une classe d’opérations non complétement continues. Annals of Math. T. 50, 140–147 (1950)

    Google Scholar 

  11. Johansson, A., Öberg, A.: Square summability of variations of g-functions and uniqueness of g-measures. Math. Res. Lett. 10 (5–6), 587–601 (2003)

    Google Scholar 

  12. Keane, M., Murray, R., Young, L.-S.: Computing invariant measures for expanding circle maps. Nonlinearity 11, 27–46 (1998)

    MATH  Google Scholar 

  13. Keller, G., Liverani, C.: Stability of the Spectrum for Transfer Operators. Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 28 (1), 141–152 (1999)

  14. Li, T.Y.: Finite approximation for the Perron–Frobenius operator. A solution to Ulam’s conjecture. J. of Approx. Theory 17, 177–186 (1976)

    MATH  Google Scholar 

  15. Öberg, A., Strichartz, R.S., Yingst, A.Q.: Level sets of harmonic functions on the Sierpinski gasket. Ark. Mat. 40 (2), 335–362 (2002)

    Google Scholar 

  16. Strichartz, R.S., Taylor, A., Zhang, T.: Densities of self-similar measures on the line. Experimental Mathematics 4, 101–128 (1995)

    MATH  Google Scholar 

  17. Viana, M.: Stochastic dynamics of deterministic systems. IMPA 1997. Available on Viana’s homepage

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Uppsala University, 480, 751 06, Uppsala, Sweden

    Anders Öberg

Authors
  1. Anders Öberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anders Öberg.

Additional information

Mathematics Subject Classification (2000): 37A30, 37C30, 37M25, 47A58

Acknowledgement I would like to express my deep gratitude to Andreas Strömbergsson and to the anonymous referee. The referee had several very enlightening comments, which Andreas helped me to deal with. Section 4 is essentially due to Andreas and he also came up with the new Proposition 3 and helped me to improve Lemma 1. Thanks also to Svante Janson, Anders Johansson, Sten Kaijser, Robert Strichartz and Hans Wallin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Öberg, A. Algorithms for approximation of invariant measures for IFS. manuscripta math. 116, 31–55 (2005). https://doi.org/10.1007/s00229-004-0515-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-004-0515-4

Keywords

Search

Navigation