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Keywords:
dense chaos; generic chaos; piecewise monotone map
dense chaos; generic chaos; piecewise monotone map
Summary:
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 $\liminf_{n\to\infty} |f^n(x)-f^n(y)|=0$ and $\limsup_{n\to\infty} |f^n(x)-f^n(y)|>0$, 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.
References:
[1] Piórek J.: On the generic chaos in dynamical systems. Acta Math. Univ. Iagell. 25 (1985), 293-298. MR 0837847
[2] Snoha L'.: Generic chaos. Comment. Math. Univ. Carolinae 31 (1990), 793-810. MR 1091377 | Zbl 0724.58044
[3] Snoha L'.: Two-parameter chaos. preprint, 1992. MR 1286993 | Zbl 0799.58051
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