Let f be a homeomorphism from the real line $\mathbb R$ to $\mathbb R$ which maps each null set ( its Lebsegue measure is zero) to a null set. The question is that : Is $f$ absolutely continuous?
If we strengthen the condition to be that $f$ is a quasisymmetric homeomorphism from $\mathbb R$ to $\mathbb R$ which maps each null set to a null set, then what is the answer?
(addition: for the detail definition of a quasisymmetric map, please see https://en.wikipedia.org/wiki/Quasisymmetric_map)
I don't believe so. For each integer $k\ge1$, define $f$ on the interval $[k,k+1)$ to be $f(x) = k+(x-k)^k$. (And define $f(x)=x$ for $x<1$.) Then $f$ is a homeomorphism which maps null sets to null sets, but $f'$ is not bounded where it's defined, hence $f$ is not absolutely continuous.