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Conformal Geometry and Dynamics

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Teichmüller space for iterated function systems


Authors: Martial R. Hille and Nina Snigireva
Journal: Conform. Geom. Dyn. 16 (2012), 132-160
MSC (2010): Primary 37F45, 37F35, 37F40, 28A80
DOI: https://doi.org/10.1090/S1088-4173-2012-00241-X
Published electronically: May 8, 2012
MathSciNet review: 2915752
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Abstract: In this paper we investigate families of iterated function systems (IFS) and conformal iterated function systems (CIFS) from a deformation point of view. Namely, we introduce the notion of Teichmüller space for finitely and infinitely generated (C)IFS and study its topological and metric properties. Firstly, we completely classify its boundary. In particular, we prove that this boundary essentially consists of inhomogeneous systems. Secondly, we equip Teichmüller space for (C)IFS with different metrics, an Euclidean, a hyperbolic, and a $ \lambda $-metric. We then study continuity of the Hausdorff dimension function and the pressure function with respect to these metrics. We also show that the hyperbolic metric and the $ \lambda $-metric induce topologies stronger than the non-metrizable $ \lambda $-topology introduced by Roy and Urbanski and, therefore, provide an alternative to the $ \lambda $-topology in the study of continuity of the Hausdorff dimension function and the pressure function. Finally, we investigate continuity properties of various limit sets associated with infinitely generated (C)IFS with respect to our metrics.


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Additional Information

Martial R. Hille
Affiliation: Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany
Email: hille@math.hu-berlin.de

Nina Snigireva
Affiliation: Mathematical Sciences Institute, John Dedman Building 27, The Australian National University, Canberra ACT 0200, Australia
Email: Nina.Snigireva@anu.edu.au

DOI: https://doi.org/10.1090/S1088-4173-2012-00241-X
Keywords: Iterated function systems, inhomogeneous iterated function systems, conformal iterated function systems, Teichmüller space, Hausdorff dimension, $𝜆$-topology.
Received by editor(s): November 29, 2011
Published electronically: May 8, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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