Non-Typical Points for $\beta $-Shifts

Volume 61 / 2013

David Färm, Tomas Persson Bulletin Polish Acad. Sci. Math. 61 (2013), 123-132 MSC: 37E05, 37C45, 11J83. DOI: 10.4064/ba61-2-5

Abstract

We study sets of non-typical points under the map $f_\beta \mapsto \beta x $ mod 1 for non-integer $\beta$ and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition $\beta>1.541$ found in the above paper can be removed.

Authors

  • David FärmInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8, P.O. Box 21
    00-956 Warszawa, Poland
    and
    Duvholmsgränd 24
    12741 Skärholmen, Sweden
    e-mail
  • Tomas PerssonInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8, P.O. Box 21
    00-956 Warszawa, Poland
    and
    Centre for Mathematical Sciences
    Lund University
    Box 118
    22100 Lund, Sweden
    e-mail

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