Real Analysis Exchange

Finer diophantine and regularity properties of 1-dimensional parabolic IFS

Mariusz Urbański

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Recall that a Borel probability measure $\mu$ on $\R$ is called extremal if $\mu$-almost every number in $\R$ is not very well approximable. In this paper, we prove extremality (and implying it the exponentially fast decay property (efd)) of conformal measures induced by $1$-dimensional finite parabolic iterated function systems. We also investigate the doubling property of these measures and we estimate from below the Hausdorff dimension of the limit sets of such iterated systems.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 143-164.

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11J99: None of the above, but in this section
Secondary: 28A78: Hausdorff and packing measures 37E05: Maps of the interval (piecewise continuous, continuous, smooth)

extremal measures parabolic IFS Hausdorff measures conformal measures not very well and badly approximable numbers extendable systems doubling property


Urbański, Mariusz. Finer diophantine and regularity properties of 1-dimensional parabolic IFS. Real Anal. Exchange 31 (2005), no. 1, 143--164.

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  • D. Kleinbock, B. Weiss, Badly Approximable Vectors on Fractals, Selecta Math., 10 (2004), 479–523.
  • D. Kleinbock, E. Lindenstrauss, B. Weiss, On Fractal Measures and Diophantine Approximation, Preprint, 2003.
  • R. D. Mauldin, M. Urbański, Parabolic Iterated Function Systems, Ergod. Th. & Dynam. Sys., 20 (2000), 1423–1447.
  • D. Mauldin, M. Urbański, Fractal Measures for Parabolic IFS, Adv. in Math., 168 (2002), 225–253.
  • D. Mauldin, M. Urbański, The Doubling Property of Conformal Measures of Infinite Iterated Function Systems, J. Numb. Theory, 102 (2003), 23–40.
  • M. Urbański, Parabolic Cantor Sets, Preprint 1995, available on Urbański's webpage.
  • M. Urbański, Parabolic Cantor Sets, Fund. Math., 151 (1996), 241–277.
  • M. Urbański, Diophantine Approximation for Conformal Measures of One-Dimensional Iterated Function Systems, Compositio Math., 141 (2005), 869–886.
  • M. Urbański, Diophantine Approximation of Self-Conformal Measures, Journal Number Th., 110 (2005), 219–235
  • B. Weiss, Almost No Points on a Cantor Set are Very Well Approximable, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 949–952.