Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

Abstract

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\mathrm{Homeo}([0,1])$ has length 2. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of $\mathrm{Homeo}([0,1])$ have length 2. We answer this question in the affirmative.

We call $f\in \mathrm{Homeo}({[0,1]}^d)$ singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of $f\in \mathrm{Homeo}([0,1])$ has length 2 iff f is singular iff f is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem.

We show that for $d\ge 2$ the graph of the generic element of $\mathrm{Homeo}({[0,1]}^d)$ has infinite d-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of $\mathrm{Homeo}({[0,1]}^d)$ with infinite d-dimensional Hausdorff measure is not Haar null. We show that for $d\ge 2$ the generic element of $\mathrm{Homeo}({[0,1]}^d)$ is singular but not strongly singular. We also show that for $d\ge 2$ almost every element of $\mathrm{Homeo}({[0,1]}^d)$ is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null).

Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.