On Lebesgue measure preserving Besicovitch functions

Abstract

We consider the space $C_λ$ of all continuous interval maps preserving the Lebesgue measure $λ$. 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 $C_λ$ is nonempty and meager. We prove that no Besicovitch function is invertible $λ$-almost everywhere. As a consequence, every Besicovitch function in $C_λ$ has positive measure-theoretic entropy with respect to $λ$. Furthermore, we show that Besicovitch functions are dense in $C_λ$ and, consequently, also dense in the class of interval maps with a dense set of periodic points.