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 $A$.
Keywords:
prove mathbb mathbb darboux has point prime period different ldots entropy positive other every set subset mathbb there almost continuous sense stallings function mathbb mathbb positive entropy which set mathop per prime periods periodic points equal
Affiliations des auteurs :
Tomasz Natkaniec
1
;
Piotr Szuca
2
1
Institute of Mathematics Gdańsk University Wita Stwosza 57 80-952 Gdańsk, Poland
2
Institute of Mathematics Gdańsk University Wita Stwosza 57 80–952 Gdańsk, Poland
@article{10_4064_cm121_1_9,
author = {Tomasz Natkaniec and Piotr Szuca},
title = {On {Pawlak's} problem concerning entropy of almost continuous
functions},
journal = {Colloquium Mathematicum},
pages = {107--111},
year = {2010},
volume = {121},
number = {1},
doi = {10.4064/cm121-1-9},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm121-1-9/}
}
TY - JOUR
AU - Tomasz Natkaniec
AU - Piotr Szuca
TI - On Pawlak's problem concerning entropy of almost continuous
functions
JO - Colloquium Mathematicum
PY - 2010
SP - 107
EP - 111
VL - 121
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4064/cm121-1-9/
DO - 10.4064/cm121-1-9
LA - en
ID - 10_4064_cm121_1_9
ER -
%0 Journal Article
%A Tomasz Natkaniec
%A Piotr Szuca
%T On Pawlak's problem concerning entropy of almost continuous
functions
%J Colloquium Mathematicum
%D 2010
%P 107-111
%V 121
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4064/cm121-1-9/
%R 10.4064/cm121-1-9
%G en
%F 10_4064_cm121_1_9
Tomasz Natkaniec; Piotr Szuca. On Pawlak's problem concerning entropy of almost continuous
functions. Colloquium Mathematicum, Tome 121 (2010) no. 1, pp. 107-111. doi: 10.4064/cm121-1-9
