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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Increasing digit subsystems of infinite iterated function systems
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by Thomas Jordan and Michał Rams PDF
Proc. Amer. Math. Soc. 140 (2012), 1267-1279 Request permission

Abstract:

We consider an infinite iterated function system $\{f_i\}_{i=1}^{\infty }$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ \cdots$ if $i_n>\Phi (i_{n-1})$ for certain increasing functions $\Phi :\mathbb N\rightarrow \mathbb N$. We compute both the Hausdorff and packing dimensions of such sets. Our results generalise work of Ramharter which shows that the set of continued fractions with strictly increasing digits has Hausdorff dimension $\frac {1}{2}$.
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Additional Information
  • Thomas Jordan
  • Affiliation: School of Mathematics, The University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, United Kingdom
  • MR Author ID: 782791
  • Email: thomas.jordan@bristol.ac.uk
  • Michał Rams
  • Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
  • MR Author ID: 656055
  • Email: M.Rams@impan.gov.pl
  • Received by editor(s): October 25, 2010
  • Received by editor(s) in revised form: December 21, 2010
  • Published electronically: July 19, 2011
  • Additional Notes: The second author’s research was supported by grants EU FP6 ToK SPADE2, EU FP6 RTN CODY and MNiSW grant ‘Chaos, fraktale i dynamika konforemna’.
  • Communicated by: Bryna Kra
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1267-1279
  • MSC (2010): Primary 28A80; Secondary 11K50
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10969-9
  • MathSciNet review: 2869111