On Pawlak's problem concerning entropy of almost continuous functions

Volume 121 / 2010

Tomasz Natkaniec, Piotr Szuca Colloquium Mathematicum 121 (2010), 107-111 MSC: Primary 26A18; Secondary 37B40, 37E05, 26A15. DOI: 10.4064/cm121-1-9

Abstract

We prove that if $f:{\mathbb I}\to{\mathbb I}$ is Darboux and has a point of prime period different from $2^i$, $i=0,1,\ldots,$ then the entropy of $f$ is positive. On the other hand, for every set $A\subset{\mathbb N}$ with $1\in A$ there is an almost continuous (in the sense of Stallings) function $f:{\mathbb I}\to{\mathbb I}$ with positive entropy for which the set $\mathop{\rm Per}(f)$ of prime periods of all periodic points is equal to $A$.

Authors

  • Tomasz NatkaniecInstitute of Mathematics
    Gdańsk University
    Wita Stwosza 57
    80-952 Gdańsk, Poland
    e-mail
  • Piotr SzucaInstitute of Mathematics
    Gdańsk University
    Wita Stwosza 57
    80–952 Gdańsk, Poland
    e-mail

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