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 | References | Similar Articles | Additional Information

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 , counting overlaps. In the case when the function 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 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:
https://doi.org/10.1090/S0002-9939-2011-10704-4

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.