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

Richárd Balka, Márton Elekes, Viktor Kiss, Márk Poór

Published 2020-09-29Version 1

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\text{Homeo}([0,1])$ 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 $\text{Homeo}([0,1])$ have length $2$. We answer this question in the affirmative. Since the graph of $f \in \text{Homeo}([0,1])$ has length $2$ iff $f$ is singular (i.e., it takes a suitable set of full measure to a nullset) iff $f$ is strongly singular (i.e., it has zero derivative almost everywhere), the following problems are all natural generalisations of Banach's observation and Mycielski's problem. What is the $d$-dimensional Hausdorff measure of the generic/almost every element of $\text{Homeo}([0,1]^d)$? Is the generic/almost every element of $\text{Homeo}([0,1]^d)$ singular? Is the generic/almost every element of $\text{Homeo}([0,1]^d)$ strongly singular? We show that for $d \ge 2$ the graph of the generic element of $\text{Homeo}([0,1]^d)$ 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 $\text{Homeo}([0,1]^d)$ with infinite $d$-dimensional Hausdorff measure is not Haar null. We show that for $d \ge 2$ the generic element of $\text{Homeo}([0,1]^d)$ is singular but not strongly singular. We also show that for $d \ge 2$ almost every element of $\text{Homeo}([0,1]^d)$ 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.