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On shrinking targets for piecewise expanding interval maps

Published online by Cambridge University Press:  25 August 2015

TOMAS PERSSON  and
MICHAŁ RAMS
TOMAS PERSSON
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, 22100 Lund, Sweden email [email protected]
MICHAŁ RAMS
Affiliation:
Instytut Matematyczny, Polska Akademia Nauk, ul. Sniadeckich 8, 00-656 Warszawa, Poland email [email protected]
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Abstract

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 646 - 663
Copyright
© Cambridge University Press, 2015 

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