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 $\log 2/2$ and generically they have infinite topological entropy. In this article we show that the generic map has zero measure-theoretic entropy. This implies that there are dramatic differences in the topological versus measure-theoretic behavior both for injectivity as well as for the structure of the level sets of generic maps. As a consequence we get a surprising corollary for a family of planar attractors homeomorphic to the pseudo-arcs.

1) The title has changed, the introduction has been reorganized and section 5 has been added, in this section we give an application to planar attractors.2) Minor changes after referee report, the article will appear in Studia Mathematica