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The Hausdorff Dimension of the Harmonic Measure on de Rham's Curve

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Abstract

De Rham's curve obtained by the “trisection” of the square can also be considered as a boundary of the random walk based on two affine maps. The aim of the present paper is to calculate the Hausdorff dimension of the harmonic measure on the curve. Bibliography: 10 titles.

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REFERENCES

  1. P. Nikitin, “Dynamic and group properties of de Rham's curve,” Diploma thesis (2001).

  2. K. Falconer, Techniques in Fractal Geometry, John Wiley Sons (1997).

  3. J. de Rham, “Une peu de mathématique a propos d'une courbe plane,” Rev. Math. Elementaires, II,Nos. 4, 5, 678–689 (1947).

    Google Scholar 

  4. J. de Rham, “Sur une courbe plane,” J. Math. Pures Appl., XXXV, 25–42 (1956).

    Google Scholar 

  5. J. de Rham, “Sur quelques courbes définies par les équations fonctionnelles,” Rendiconti del Seminario Matematico dell'Università e del Politechnico di Torino, XVI, 101–113 (1957).

    Google Scholar 

  6. J. de Rham, “Sur les courbes limites de poligones obtenu par trisection,” L'Enseignement Mathématique, II-ième Série, V, 29–43 (1959).

    Google Scholar 

  7. J. E. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math J., 30, 713–747 (1981).

    Google Scholar 

  8. N. Sidorov and A. Vershik, “Ergodic properties of Erdos measure, the entropy of goldenshift, and related problems,” Monatsh. Math., 126, 215–261 (1998).

    Google Scholar 

  9. M. Nikol, N. Sidorov and D. Broomhead, “On the fine structure of stationary measures in systems which contracts on average,” J. Theoret. Prob., 15, 715–730 (2002).

    Google Scholar 

  10. S.-L. Young, “Dimension, entropy and Lyapunov exponents,” Ergod. Theory Dynam. Sys., 2, 109–124 (1982).

    Google Scholar 

  11. A. Ya. Khinchin, Continued fractions [in Russian], Moscow (1978).

  12. I. A. Ibragimov and Yu. V. Linnik, Indepemdent and Stationary Correlated Variables [in Russian], Moscow (1965).

  13. P. Billingsley, Ergodic Theory and Information, New York (1965).

  14. V. Kaimanovich and A. Vershik, “Random walks on discrete group: boundary and entropy,” Ann. Prob., 11, 457–490 (1983).

    Google Scholar 

  15. R. Salem, “On some singular monotonic functions which are strictly increasing,” Trans. Amer. Math. Soc., 53, 427–439 (1943).

    Google Scholar 

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  1. St.Petersburg Department of the Steklov Mathematical Institute, Russia

    P. P. Nikitin

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  1. P. P. Nikitin
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Nikitin, P.P. The Hausdorff Dimension of the Harmonic Measure on de Rham's Curve. Journal of Mathematical Sciences 121, 2409–2418 (2004). https://doi.org/10.1023/B:JOTH.0000024622.17820.37

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  • DOI: https://doi.org/10.1023/B:JOTH.0000024622.17820.37

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