Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e.

Abstract: We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this article we show that thegeneric map has zero measure-theoreticentropy. This implies that there are dramatic differences inthe topological versus measure-theoretic behavior both for injectivity as well as for the structure of thelevel sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arcs.