There exists no borelian classification of Brouwer homeomorphisms

In this paper, we consider the space B-2 of the planar homeomorphisms that are obtained by gluing two translations together, endowed with the compact- open topology. We construct an embedding of the Canter set {0 ,1}(Z) in B-2 , such that in this Canter set of homeomorphisms the conjugacy classes are given by the orbits of the shift map on {0, 1)(Z) As a classical consequenc e, we explain why there cannot be any borelian classification of the conjug acy relation on the space B-2.