There is no Borelian classification of the Brouwer homeomorphisms (Q2709597)

An orientation-preserving homeomorphism of the plane without fixed point is called a Brouwer homeomorphism. For such a homeomorphism each point of the plane belongs to a translation domain, i.e. to an open simply connected invariant domain, on which the homeomorphism is conjugated to a translation. The author constructs a set \({\mathcal B}_2\) of Brouwer homeomorphisms by glueing two translations together. There is an embedding of a Cantor set in \({\mathcal B}_2\), such that in this Cantor set the conjugacy classes are given by orbits of the shift map on \(\{0,1\}^ {\mathbb Z}\). It follows that there is not any Borelian classification of the conjugacy relation in \({\mathcal B}_2\).