Abstract
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.
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Partially supported by grant BW/5100-5-0201-6.
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Nowik, A. A Vitali set can be homeomorphic to its complement. Acta Math Hung 115, 145–154 (2007). https://doi.org/10.1007/s10474-007-5222-7
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DOI: https://doi.org/10.1007/s10474-007-5222-7