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arXiv · 2602.19800

On Lebesgue measure preserving Besicovitch functions

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

We consider the space $C_{\lambda}$ of all continuous interval maps preserving the Lebesgue measure $\lambda$. A continuous function $f\colon~[0,1]\to \mathbb R$ is called Besicovitch if it does not have any finite or infinite unilateral derivative. It is known that the set of Besicovitch functions in $C_{\lambda}$ is nonempty and meager. We prove that no Besicovitch function is invertible $\lambda$-almost everywhere. As a consequence, every Besicovitch function in $C_{\lambda}$ has positive measure-theoretic entropy with respect to $\lambda$. Furthermore, we show that Besicovitch functions are dense in $C_{\lambda}$ and, consequently, also dense in the class of interval maps with a dense set of periodic points.

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Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy. 2026-02-23. On Lebesgue measure preserving Besicovitch functions. https://arxiv.org/abs/2602.19800

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