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Positive operators and Hausdorff dimension of invariant sets

Authors: Roger D. Nussbaum, Amit Priyadarshi and Sjoerd Verduyn Lunel
Journal: Trans. Amer. Math. Soc. 364 (2012), 1029-1066
MSC (2010): Primary 37F35, 28A80; Secondary 37C30, 47B65
Published electronically: October 5, 2011
MathSciNet review: 2846362
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Abstract: In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are ``infinitesimal similitudes'' on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of $ \sigma > 0$ for which $ r(L_\sigma )=1$, where $ L_\sigma $, $ \sigma >0$, is a naturally associated family of positive linear operators and $ r(L_\sigma )$ denotes the spectral radius of $ L_\sigma $. We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

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Additional Information

Roger D. Nussbaum
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Amit Priyadarshi
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Sjoerd Verduyn Lunel
Affiliation: Mathematisch Instituut, Universiteit Leiden, Leiden, The Netherlands

Keywords: Hausdorff dimension, iterated function systems, positive operators, spectral radius
Received by editor(s): February 1, 2010
Received by editor(s) in revised form: October 5, 2010
Published electronically: October 5, 2011
Additional Notes: The first author was supported in part by NSF Grant DMS-0701171.
Article copyright: © Copyright 2011 American Mathematical Society

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