We prove that there is a homeomorphism of the unit interval onto itself that is so singular that it maps some set $E$ of $\dim_HE=0$ onto a set $F$ of $\dim_H[0,1]\setminus F=0$.
WEI, CHUN and WEN, SHENGYOU
"A remark on singularity of homeomorphisms and Hausdorff dimension,"
Turkish Journal of Mathematics: Vol. 40:
4, Article 7.
Available at: https://journals.tubitak.gov.tr/math/vol40/iss4/7