A Vitali set can be homeomorphic to its complement

We prove that there exists a special homeomorphism of the Cantor space such that every noncancellable composition of finite powers and translations of rational numbers has no fixed point. For this homeomorphism there exists both a Vitali and Bernstein subset of the Cantor set such that the image of this set is equal to its complement. There exists a Bernstein and Vitali set such that there is no Borel isomorphism between this set and its complement. Partially supported by grant BW/5100-5-0201-6.