For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $limin{f}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and $limsu{p}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$. We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

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 ${lim\; inf}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|=0$ and ${lim\; sup}_{n\to \infty }|f^n\left(x\right)-f^n\left(y\right)|>0$, 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...