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Invariant measures for parabolic IFS with overlaps and random continued fractions


Authors: K. Simon, B. Solomyak and M. Urbanski
Journal: Trans. Amer. Math. Soc. 353 (2001), 5145-5164
MSC (2000): Primary 37L30; Secondary 60G30
DOI: https://doi.org/10.1090/S0002-9947-01-02873-2
Published electronically: July 12, 2001
MathSciNet review: 1852098
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Abstract:

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.'' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.


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Additional Information

K. Simon
Affiliation: Department of Stochastics, Institute of Mathematics, Technical University of Budapest, P.O. Box 91, 1521 Budapest, Hungary
Email: simonk@math.bme.hu

B. Solomyak
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195-4350
Email: solomyak@math.washington.edu

M. Urbanski
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203-1430
Email: urbanski@unt.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02873-2
Keywords: Iterated function systems, parabolic maps, random continued fractions
Received by editor(s): January 17, 2000
Received by editor(s) in revised form: December 18, 2000
Published electronically: July 12, 2001
Additional Notes: Research of Simon was supported in part by the OTKA foundation grant F019099. Research of Solomyak was supported in part by the Fulbright foundation and the NSF grant DMS 9800786. Research of Urbański was supported in part by the NSF grant DMS 9801583
Article copyright: © Copyright 2001 American Mathematical Society

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