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 all such maps have topological entropy at least 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.
Keywords
Cite
@article{arxiv.2405.09917,
title = {Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e},
author = {Jozef Bobok and Jernej Činč and Piotr Oprocha and Serge Troubetzkoy},
journal= {arXiv preprint arXiv:2405.09917},
year = {2026}
}
Comments
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