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

TL;DR

This paper studies continuous Lebesgue measure-preserving interval maps, revealing that generically they are nowhere monotone, invertible almost everywhere, and exhibit complex entropy behavior with significant differences between topological and measure-theoretic properties.

Contribution

It demonstrates that generic measure-preserving maps are nowhere monotone, invertible almost everywhere, and have zero measure-theoretic entropy despite high topological entropy.

Findings

Generic maps have infinite topological entropy.

Most maps are nowhere monotone.

Generic maps have zero measure-theoretic entropy.

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 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.