arXiv subjects
Publications and source records attributed to Marshall Williams.
A standard introductory result is that Hausdorff spaces have the property US, that is, each convergent sequence has a unique limit. This paper explores several existing and new characterizations of separation axioms that are strictly weaker than $T_2$ but strictly stronger than US.
Assume that $X$ and $Y$ are locally compact and locally doubling metric spaces, which are also generalized $n$-manifolds, that $X$ is locally linearly locally $n$-connected, and that $Y$ has bounded turning. In this paper, addressing Heinonen's ICM 02 talk, we study the geometry of the branch set $\mathcal{B}_f$ of a quasiregular mapping between metric $n$-manifolds. In particular, we show that $\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}$ is countably porous, as is its image $f\big(\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}\big)$. As a corollary, $\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}$ and its image are null sets with respect to any locally doubling measures on $X$ and Y, respectively. Moreover, if either $H_f(x)\leq H$ or $H_f^*(x)\leq H^*$ for all $x\in X$, then both $\mathcal{B}_f$ and $f\big(\mathcal{B}_f\big)$ are countably $δ$-porous, quantitatively, with a computable porosity constant. When further metric and analytic assumptions are placed on $X$, $Y$, and $f$, our theorems generalize the well-known Bonk--Heinonen theorem and Sarvas' theorem to a large class of metric spaces. Moreover, our results are optimal in terms of the underlying geometric structures. As a direct application, we obtain the important Väisälä's inequality in greatest generality. Applying our main results to special cases, we solve an open problem of Heinonen--Rickman and an open question of Heinonen--Semmes.
In this paper, we develop the foundations of the theory of quasiregular mappings in general metric measure spaces. In particular, nine definitions of quasiregularity for a discrete open mapping with locally bounded multiplicity are proved to be quantitatively equivalent when the metric measure spaces have locally bounded geometry. We also demonstrate that some, though not all, of these implications remain true under fairly general hypotheses. The major new tool appeared in our approach is a powerful factorization, termed the pullback factorization, of a quasiregular mapping into the composition of a 1-BLD mapping and a quasiconformal mapping in locally compact complete metric measure spaces. This factorization also brings fundamental new point of view of the theory of quasiregular mappings in Euclidean spaces, in particular, a branched counterpart of quasisymmetric mappings is introduced and is shown to be locally equivalent with quasiregular mappings, quantitatively. As applications of our new techniques, we answer some well-known open problems in this field and characterize BLD mappings in metric spaces with locally bounded geometry. In particular, a conjecture of Heinonen--Rickman is shown to be false and an open problem of Heinonen--Rickman gets solved affirmatively.
We show that if $f:X\to Y$ is a quasisymmetric mapping between Ahlfors regular spaces, then $\dim_H f(E)\leq\dim_H E$ for "almost every" bounded Ahlfors regular set $E\subseteq X$. If additionally, $X$ and $Y$ are Loewner spaces then $\dim_H f(E)=\dim_H E$ for "almost every" Ahlfors regular set $E\subset X$. The precise statements of these results are given in terms of Fuglede's modulus of measures. As a corollary of these general theorems we show that if $f$ is a quasiconformal map of $\mathbb{R}^N$, $N\geq 2$, then for Lebesgue a.e. $y\in\mathbb{R}^N$ we have $\dim_H f(y+E) = \dim_H E$. A similar result holds for Carnot groups as well. For planar quasiconformal maps, our general estimates imply that if $E \subset \mathbb{R}$ is Ahlfors $d$-regular, $d<1$, then some component of $f(E \times \mathbb{R})$ has dimension at most $2/(d+1)$, and we construct examples to show this bound is sharp. In addition, we show there is a $1$-dimensional set $S\subseteq \mathbb R$ and planar quasiconformal map $f$ such that $f(\mathbb{R} \times S)$ contains no rectifiable sub-arcs. These results generalize work of Balogh, Monti and Tyson \cite{Tyson:frequency} and answer questions posed in \cite{Tyson:frequency} and \cite{AimPL}.
In this paper, we provide an alternative appraoch to an expectaion of F\"assler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis-Mostow, Karmanova and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam and Tyson to establish the necessary analytic foundations.
We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either space. When $X$ and $Y$ have locally $Q$-bounded geometry and $Y$ is contained in an Alexandrov space of curvature bounded above, the sharpness of our results implies that, as in the classical case, the modular and pointwise outer dilatations of $\map$ are related by $K_O(f)= \operatorname{esssup} H_O(x,f)$.
We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we deduce a generalization of the chain rule, and show that currents of absolutely continuous mass are given by integration against measurable $k$-vector fields. We further prove that if the underlying metric space is a Carnot group with its Carnot-Carathéodory distance, then every metric current $T$ satisfies $T\lfloor_θ=0$ and $T\lfloor_{dθ}=0$, whenever $θ\in Ω^{1}(\mathbb{G})$ annihilates the horizontal bundle of $\mathbb G$. Moreover, this condition is necessary and sufficient for a metric current with respect to the Riemannian metric to extend to one with respect to the Carnot-Carathéodory metric, provided the current either is locally normal, or has absolutely continuous mass.