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Jeremy T. Tyson

Publications and source records attributed to Jeremy T. Tyson.

At least 19 recordsLinked to original sources
arXiv2026

Quasiconformal and Sobolev distortion of dimension

We review a selection of the literature on the distortion of metric notions of dimension under quasiconformal, quasisymmetric, and Sobolev mappings. Our story begins with Gehring's landmark 1973 higher integrability theorem for quasiconformal maps, along with its implications for the distortion of Hausdorff dimension. Astala's 1994 solution to the planar higher integrability conjecture led to renewed interest in the subject in two dimensions. We continue with results from the 2000s and 2010s on the distortion of dimension by Sobolev maps, including estimates for dimension increase for generic elements in parameterized families of subsets. In the abstract metric setting, Pansu's notion of conformal dimension provides a key quasisymmetric invariant which has been useful in a wide range of applications. We briefly review relevant facts about conformal dimension, highlighting results of interest in the Euclidean setting. We conclude with recent work of the author in collaboration with Chrontsios Garitsis and with Fraser, extending the previous theory to interpolating dimensions and providing new insight into both quasiconformal classification and conformal dimension.

math.MG↗
arXiv2026

Analytic and quasiregular distortion of Nagata dimension

We study how analytic functions, and more generally quasiregular mappings, distort Nagata dimension. Quasiconformal mappings of domains preserve the Nagata dimension of compact subsets, in view of a result of Lang and Schlichenmaier. We establish the same conclusion for analytic functions defined on general planar domains. On the other hand, polynomials (and more generally, rational maps) preserve the Nagata dimension of arbitrary subsets of their domain. In the absence of the compactness assumption, we provide examples to show that an entire function can increase or decrease the Nagata dimension of subsets of the domain. Some of these results generalize to meromorphic functions, and separately to planar quasiregular maps in view of Stoilow factorization. We also show that conformal mappings can change the porosity behavior of noncompact subsets of their domain; this yields examples of planar conformal maps which take sets of Nagata dimension strictly less than two onto set of Nagata dimension two. We conclude with open questions and potential future work related to the distortion of Nagata dimension by higher-dimensional quasiregular maps.

math.CV↗
arXiv2025

Sobolev and quasiconformal distortion of intermediate dimension with applications to conformal dimension

We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the Gehring--Väisälä theorem on dilatation-dependent quasiconformal distortion of dimension and Kovalev's theorem on the nonexistence of metric spaces with conformal dimension strictly between zero and one. Applications include new contributions to the quasiconformal classification of Euclidean sets and a new sufficient condition for the vanishing of conformal box-counting dimension. We illustrate our conclusions with specific consequences for Bedford--McMullen carpets, samples of Mandelbrot percolation, and product sets containing a polynomially convergent sequence factor.

math.MG↗
arXiv2025

The Heinonen-Semmes problems after thirty years

We survey the current status of the questions posed by Juha Heinonen and Stephen Semmes in `Thirty-three yes or no questions about mappings, measures, and metrics' (Conformal Geometry and Dynamics, 1997).

math.MG↗
arXiv2025

On the Assouad spectrum of Hölder and Sobolev graphs

We provide upper bounds for the Assouad spectrum $\dim_A^θ(\text{Gr}(f))$ of the graph of a real-valued Hölder or Sobolev function $f$ defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $α$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $α$ and returns the graph of a new $α$-Hölder continuous function for which the Assouad $θ$-spectrum realizes the stated upper bound for all $θ\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

math.CA↗
arXiv2025

Metric spaces with small rough angles and the rectifiability of rough self-contracting curves

The small rough angle ($\mbox{SRA}$) condition, introduced by Zolotov in arXiv:1804.00234, captures the idea that all angles formed by triples of points in a metric space are small. In the first part of the paper, we develop the theory of metric spaces $(X,d)$ satisfying the $\mbox{SRA}(α)$ condition for some $α<1$. Given a metric space $(X,d)$ and $0<α<1$, the space $(X,d^α)$ satisfies the $\mbox{SRA}(2^α-1)$ condition. We prove a quantitative converse up to bi-Lipschitz change of the metric. We also consider metric spaces which are $\mbox{SRA}(α)$ free (there exists a uniform upper bound on the cardinality of any $\mbox{SRA}(α)$ subset) or $\mbox{SRA}(α)$ full (there exists an infinite $\mbox{SRA}(α)$ subset). Examples of SRA free spaces include Euclidean spaces, finite-dimensional Alexandrov spaces of non-negative curvature, and Cayley graphs of virtually abelian groups; examples of $\mbox{SRA}$ full spaces include the sub-Riemannian Heisenberg group, Laakso graphs, and Hilbert space. We study the existence or nonexistence of $\mbox{SRA}(ε)$ subsets for $0<ε<2^α-1$ in metric spaces $(X,d^α)$ for $0<α<1$. In the second part of the paper, we apply the theory of metric spaces with small rough angles to study the rectifiability of roughly self-contracting curves. In the Euclidean setting, this question was studied by Daniilidis, Deville, and the first author using direct geometric methods. We show that in any $\mbox{SRA}(α)$ free metric space $(X,d)$, there exists $λ_0 = λ_0(α)>0$ so that any bounded roughly $λ$-self-contracting curve in $X$, $λ\le λ_0$, is rectifiable. The proof is a generalization and extension of an argument due to Zolotov, who treated the case $λ=0$, i.e., the rectifiability of self-contracting curves in $\mbox{SRA}$ free spaces.

math.MG↗
arXiv2024

On the H-type deviation of step two Carnot groups

The H-type deviation, $δ({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an earlier paper, the author defined this notion and used it to provide new analytic characterizations for the class of H-type groups. In addition, a quantitative conjecture relating the H-type deviation to the behavior of the $\infty$-Laplacian of Folland's fundamental solution for the $2$-Laplacian was formulated; an affirmative answer to this conjecture would imply that all step two polarizable groups are of H-type. In this paper, we elucidate further properties of the H-type deviation. First, we show that $0\le δ({\mathbb G}) \le 1$ for all step two Carnot groups ${\mathbb G}$. Recalling that $δ({\mathbb F}_{2,m}) = \sqrt{(m-2)/m}$, where ${\mathbb F}_{2,m}$ is the free step two Carnot group of rank $m$, we conjecture that $δ({\mathbb G}) \le \sqrt{(m-2)/m}$ for all step two rank $m$ groups. We explicitly compute $δ({\mathbb G})$ when ${\mathbb G}$ is a product of Heisenberg groups and verify the conjectural upper bound for such groups, with equality if and only if ${\mathbb G}$ factors over the first Heisenberg group. We also prove the following rigidity statement: for each $m \ge 3$ there exists $δ_0(m)>0$ so that if ${\mathbb G}$ is a step two and rank $m$ Carnot group with $δ({\mathbb G}) < δ_0(m)$, then ${\mathbb G}$ enjoys certain algebraic properties characteristic of H-type groups.

math.DG↗
arXiv2022

Polar Coordinates in Carnot groups II

A Carnot group is polarizable if it carries a homogeneous norm whose powers are fundamental solutions for the $p$-sub-Laplacian operators for all $1<p \le \infty$. Such groups also support a system of horizontal polar coordinates. We prove that the converse statement is true: if a Carnot group supports a horizontal polar coordinate system with suitable properties, then it is polarizable.

math.AP↗
arXiv2022

Stability theorems for H-type Carnot groups

We introduce the H-type deviation $δ({\mathbb G})$ of a step two Carnot group ${\mathbb G}$, which measures the deviation of the group from the class of Heisenberg-type groups. We show that $δ({\mathbb G})=0$ if and only if ${\mathbb G}$ carries a vertical metric which endows it with the structure of an H-type group. We compute the H-type deviation for several naturally occurring families of step two groups. In addition, we provide analytic expressions which are comparable to the H-type deviation. As a consequence, we establish new analytic characterizations for the class of H-type groups. For instance, denoting by $N(g)=(||x||_h^4+16||t||_v^2)^{1/4}$, $g=\exp(x+t)$, the canonical Kaplan-type quasi-norm in a step two group ${\mathbb G}$ with taming Riemannian metric $g_h\oplus g_v$, we show that ${\mathbb G}$ is H-type if and only if $||\nabla_0 N(g)||_h^2=||x||_h^2/N(g)^2$ for all $g\ne 0$. Similarly, we show that ${\mathbb G}$ is H-type if and only if $N^{2-Q}$ is ${\mathcal L}$-harmonic in ${\mathbb G} \setminus \{0\}$. Here $\nabla_0$ denotes the horizontal differential operator, ${\mathcal L}$ the canonical sub-Laplacian, and $Q = \dim{\mathfrak v}_1+2\dim{\mathfrak v}_2$ the homogeneous dimension of ${\mathbb G}$, where ${\mathfrak v}_1\oplus{\mathfrak v}_2$ is the stratification of the Lie algebra. It is well-known that H-type groups satisfy both of these analytic conclusions. The new content of these results lies in the converse directions. Motivation for this work comes from a longstanding conjecture regarding polarizable Carnot groups. We formulate a quantitative stability conjecture regarding the fundamental solution for the sub-Laplacian on step two Carnot groups. Its validity would imply that all step two polarizable groups admit an H-type group structure. We confirm this conjecture for a sequence of anisotropic Heisenberg groups.

math.DG↗
arXiv2022

Quasiconformal distortion of the Assouad spectrum and classification of polynomial spirals

We investigate the distortion of Assouad dimension and the Assouad spectrum under Euclidean quasiconformal maps. Our results complement existing conclusions for Hausdorff and box-counting dimension due to Gehring--Väisälä and others. As an application, we classify polynomial spirals $S_a:=\{x^{-a}e^{\mathbf{i} x}:x>0\}$ up to quasiconformal equivalence, up to the level of the dilatation. Specifically, for $a>b>0$ we show that there exists a quasiconformal map $f$ of $\mathbb{C}$ with dilatation $K_f$ and $f(S_a)=S_b$ if and only if $K_f \ge \tfrac{a}{b}$.

math.CV↗
arXiv2019

Analysis on Metric Spaces

This note is a survey of Analysis on Metric spaces, in connection with the upcoming AMS Mathematics Research Communities program in June 2020.

math.CV↗
arXiv2018

On uniform measures in the Heisenberg group

We initiate a classification of uniform measures in the first Heisenberg group $\mathbb H$ equipped with the Korányi metric $d_H$, that represents the first example of a noncommutative stratified group equipped with a homogeneous distance. We prove that $1$-uniform measures are proportional to the spherical $1$-Hausdorff measure restricted to an affine horizontal line, while $2$-uniform measures are proportional to spherical $2$-Hausdorff measure restricted to an affine vertical line. It remains an open question whether $3$-uniform measures are proportional to the restriction of spherical $3$-Hausdorff measure to an affine vertical plane. We establish this conclusion in case the support of the measure is a vertically ruled surface. Along the way, we derive asymptotic formulas for the measures of small extrinsic balls in $({\mathbb H},d_H)$ intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in $\mathbb H$.

math.MG↗
arXiv2017

Cantor set arithmetic

Every element $u$ of $[0,1]$ can be written in the form $u=x^2y$, where $x,y$ are elements of the Cantor set $C$. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand the set of real numbers $v$ that can be written in the form $v=xy$ with $x$ and $y$ in $C$ is a closed subset of $[0,1]$ with Lebesgue measure strictly between $\tfrac{17}{21}$ and $\tfrac89$. We also describe the structure of the quotient of $C$ by itself, that is, the image of $C\times (C \setminus \{0\})$ under the function $f(x,y) = x/y$.

math.MG↗
arXiv2017

Quasiconvexity in the Heisenberg group

We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have quasiconvex complements. Conversely, we exhibit a compact totally disconnected set of Hausdorff dimension three whose complement is not quasiconvex.

math.MG↗
arXiv2016

Heisenberg quasiregular ellipticity

Following the Euclidean results of Varopoulos and Pankka--Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold $M$ to admit a nonconstant quasiregular mapping from the sub-Riemannian Heisenberg group $\mathbb{H}$. As an application, we show that a link complement $S^3\backslash L$ has a sub-Riemannian metric admitting such a mapping only if $L$ is empty, the unknot or Hopf link. In the converse direction, if $L$ is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from $\mathbb{H}$ to $S^3\backslash L$. The main result is obtained by translating a growth condition on $π_1(M)$ into the existence of a supersolution to the $4$-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces.

math.GT↗
arXiv2016

Superposition of fundamental solutions of second order quasilinear equations

We prove a superposition principle in the spirit of Crandall-Zhang and Lindqvist-Manfredi for a class of second order quasilinear equations. Riesz potentials of nonnegative and compactly supported continuous functions are either subsolutions or supersolutions for the operators associated to the stationary form of the doubly nonlinear diffusion equation. This class of operators includes both the p-Laplacian as well as the stationary porous medium equation.

math.AP↗
arXiv2016

Conformal graph directed Markov systems on Carnot groups

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

math.DS↗