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Increasing digit subsystems of infinite iterated function systems

Authors: Thomas Jordan and Michał Rams
Journal: Proc. Amer. Math. Soc. 140 (2012), 1267-1279
MSC (2010): Primary 28A80; Secondary 11K50
Published electronically: July 19, 2011
MathSciNet review: 2869111
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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

Michał Rams
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
MR Author ID: 656055

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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.