Singularity of typical homeomorphisms of the d-dimensional cube

Abstract: We establish results on the singularity of the prevalent (in the sense of Christensen) and generic (in the sense of Baire category) element of Homeo([0,1]^d). In particular, we answer a question of J. Mycielski by showing that the graph of the prevalent element of Homeo([0,1]) has length 2. We also investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them. This is a joint work with M. Elekes, V. Kiss, and M. Poór.