On Lebesgue measure preserving Besicovitch functions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908848648880128 |
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| author | Bobok, Jozef Činč, Jernej Oprocha, Piotr Troubetzkoy, Serge |
| author_facet | Bobok, Jozef Činč, Jernej Oprocha, Piotr Troubetzkoy, Serge |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19800 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Lebesgue measure preserving Besicovitch functions Bobok, Jozef Činč, Jernej Oprocha, Piotr Troubetzkoy, Serge Dynamical Systems 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. |
| title | On Lebesgue measure preserving Besicovitch functions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2602.19800 |