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Generic point equivalence and Pisot numbers

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Low-dimensional dynamical systems

Published online by Cambridge University Press:  11 July 2019

SHIGEKI AKIYAMA Open the ORCID record for SHIGEKI AKIYAMA [Opens in a new window] ,
HAJIME KANEKO  and
DONG HAN KIM Open the ORCID record for DONG HAN KIM [Opens in a new window]
SHIGEKI AKIYAMA
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8571, Japan email akiyama@math.tsukuba.ac.jp, kanekoha@math.tsukuba.ac.jp
HAJIME KANEKO
Affiliation:
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki305-8571, Japan email akiyama@math.tsukuba.ac.jp, kanekoha@math.tsukuba.ac.jp
DONG HAN KIM
Affiliation:
Department of Mathematics Education, Dongguk University – Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul04620, Korea email kim2010@dongguk.edu
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Abstract

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.

Keywords

normal numberbeta expansiongeneric pointPisot number

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3169 - 3180
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Copyright
© Cambridge University Press, 2019

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