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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Positive operators and Hausdorff dimension of invariant sets
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by Roger D. Nussbaum, Amit Priyadarshi and Sjoerd Verduyn Lunel PDF
Trans. Amer. Math. Soc. 364 (2012), 1029-1066 Request permission

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
  • MR Author ID: 132680
  • Email: nussbaum@math.rutgers.edu
  • Amit Priyadarshi
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • Email: amitpriy@math.rutgers.edu
  • Sjoerd Verduyn Lunel
  • Affiliation: Mathematisch Instituut, Universiteit Leiden, Leiden, The Netherlands
  • Email: verduyn@math.leidenuniv.nl
  • 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.
  • © Copyright 2011 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 1029-1066
  • MSC (2010): Primary 37F35, 28A80; Secondary 37C30, 47B65
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05484-X
  • MathSciNet review: 2846362