Published online by Cambridge University Press: 04 October 2017
We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which
$x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at
$x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of
$x_{0}$.