Dense chaos

Snoha, Ľubomír. "Dense chaos." Commentationes Mathematicae Universitatis Carolinae 33.4 (1992): 747-752. <http://eudml.org/doc/247389>.

@article{Snoha1992,
abstract = {According to A. Lasota, a continuous function $f$ from a real compact interval $I$ into itself is called generically chaotic if the set of all points $(x,y)$, for which $\liminf _\{n\rightarrow \infty \} |f^n(x)-f^n(y)|=0$ and $\limsup _\{n\rightarrow \infty \} |f^n(x)-f^n(y)|>0$, is residual in $I\times I$. Being inspired by this definition we say that $f$ is densely chaotic if this set is dense in $I\times I$. A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number of pieces the notion of dense chaos and that of generic chaos coincide.},
author = {Snoha, Ľubomír},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {dense chaos; generic chaos; piecewise monotone map; piecewise monotone maps; dense chaos; generic chaos},
language = {eng},
number = {4},
pages = {747-752},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Dense chaos},
url = {http://eudml.org/doc/247389},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Snoha, Ľubomír
TI - Dense chaos
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 4
SP - 747
EP - 752
AB - According to A. Lasota, a continuous function $f$ from a real compact interval $I$ into itself is called generically chaotic if the set of all points $(x,y)$, for which $\liminf _{n\rightarrow \infty } |f^n(x)-f^n(y)|=0$ and $\limsup _{n\rightarrow \infty } |f^n(x)-f^n(y)|>0$, is residual in $I\times I$. Being inspired by this definition we say that $f$ is densely chaotic if this set is dense in $I\times I$. A characterization of the generically chaotic functions is known. In the paper the densely chaotic functions are characterized and it is proved that in the class of piecewise monotone maps with finite number of pieces the notion of dense chaos and that of generic chaos coincide.
LA - eng
KW - dense chaos; generic chaos; piecewise monotone map; piecewise monotone maps; dense chaos; generic chaos
UR - http://eudml.org/doc/247389
ER -