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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hausdorff dimension of the limit set of conformal iterated function systems with overlaps


Authors: Eugen Mihailescu and Mariusz Urbański
Journal: Proc. Amer. Math. Soc. 139 (2011), 2767-2775
MSC (2010): Primary 37C45, 37D35, 28A80
Published electronically: January 6, 2011
MathSciNet review: 2801617
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Abstract: We give a new approach to the study of conformal iterated function systems with arbitrary overlaps. We provide lower and upper estimates for the Hausdorff dimension of the limit sets of such systems; these are expressed in terms of the topological pressure and the function $ d$, counting overlaps. In the case when the function $ d$ is constant, we get an exact formula for the Hausdorff dimension. We also prove that in certain cases this formula holds if and only if the function $ d$ is constant.


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Additional Information

Eugen Mihailescu
Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P. O. Box 1-764, RO 014700, Bucharest, Romania
Email: Eugen.Mihailescu@imar.ro

Mariusz Urbański
Affiliation: Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, Texas 76203-1430
Email: urbanski@unt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10704-4
PII: S 0002-9939(2011)10704-4
Keywords: Conformal iterated function systems, Hausdorff dimension, overlaps, equilibrium states
Received by editor(s): February 9, 2010
Received by editor(s) in revised form: July 17, 2010
Published electronically: January 6, 2011
Additional Notes: The research of the first author supported by CNCSIS-UEFISCSU through project PN II IDEI-1191/2008.
The research of the second author supported in part by NSF grant DMS 0700831.
Communicated by: Bryna Kra
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.