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

TL;DR

This paper investigates the measure-theoretic and topological properties of homeomorphisms on multi-dimensional cubes, revealing that generic maps are singular with infinite Hausdorff measure, contrasting with the one-dimensional case.

Contribution

It extends Banach's one-dimensional results to higher dimensions, showing generic homeomorphisms are singular with infinite Hausdorff measure and analyzing their measure-theoretic properties.

Findings

Generic homeomorphisms in higher dimensions have infinite Hausdorff measure.

Almost every homeomorphism is singular but not strongly singular.

The set of strongly singular maps is Haar ambivalent.

Abstract

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. We call $f \in \text{Homeo}([0,1]^d)$ 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 $f \in \text{Homeo}([0,1])$ 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 $\text{Homeo}([0,1]^d)$ 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.