# A Besicovitch-Morse function preserving the Lebesgue measure

## TL;DR

This paper constructs a Lebesgue measure-preserving Besicovitch-Morse function within ergodic theory and shows that such functions are topologically rare among continuous measure-preserving functions.

## Contribution

It introduces a new example of a measure-preserving Besicovitch-Morse function and analyzes its topological properties within the space of continuous measure-preserving functions.

## Key findings

- Constructed a Lebesgue measure-preserving Besicovitch-Morse function.
- Proved the set of such functions is of first category among continuous measure-preserving functions.

## Abstract

We continue the investigation of which non-dierentiable maps can occur in the framework of ergodic theory started in [2]. We construct a Besicovitch-Morse function map which preserves the Lebesgue measure. We also show that the set of Besicovitch functions is of rst category in the set of continuous functions which preserve the Lebesgue measure.