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Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems

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Generic point equivalence and Pisot numbers

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

Published online by Cambridge University Press:  11 July 2019

SHIGEKI AKIYAMA Open the ORCID record for SHIGEKI AKIYAMA in new tab/window[Opens in a new window] ,
HAJIME KANEKO  and
DONG HAN KIM Open the ORCID record for DONG HAN KIM in new tab/window[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
Corresponding
E-mail address:
akiyama@math.tsukuba.ac.jp
kanekoha@math.tsukuba.ac.jp
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 number beta expansion generic point Pisot 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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