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SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory Complex dynamical systems

Published online by Cambridge University Press:  02 June 2015

LIOR FISHMAN ,
BILL MANCE ,
DAVID SIMMONS  and
MARIUSZ URBAŃSKI
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email lior.fishman@unt.edu
BILL MANCE
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email mance@unt.edu
DAVID SIMMONS*
Affiliation:
Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH 43210-1174, USA email simmons.465@osu.edu
MARIUSZ URBAŃSKI
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email urbanski@unt.edu
*
✉simmons.465@osu.edu
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Abstract

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We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

Keywords

nonautonomous dynamical systemCantor series expansionHausdorff dimensionshrinking target schemeDiophantine approximation

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 37A50: Relations with probability theory and stochastic processes 37F35: Conformal densities and Hausdorff dimension
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 205 - 213
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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