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arXiv · 2405.09917

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

Jozef Bobok·Jernej Činč·Piotr Oprocha·Serge Troubetzkoy
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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.

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Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy. 2024-05-16. Generic continuous Lebesgue measure-preserving interval maps are nowhere monotone but invertible a.e. https://arxiv.org/abs/2405.09917

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