Related papers: Analytic and quasiregular distortion of Nagata dim…
We investigate the distortion of the Assouad dimension and (regularized) spectrum of sets under planar quasiregular maps. While the respective results for the Hausdorff and upper box-counting dimension follow immediately from their…
We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions. Our main…
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 in the plane are orientation preserving homeomorphisms that satisfy certain distortion inequalities; infinitesimally, they map circles to ellipses of bounded eccentricity. Such maps have many useful geometric distortion…
We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a…
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as…
We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically…
We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian…
We show that for any dimension t>2(1+alpha K)/(1+K) there exists a compact set E of dimension t and a function alpha-Holder continuous on the plane, which is K-quasiregular only outside of E. To do this, we construct an explicit…
We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our…
We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with…
We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for…
We consider a strengthening of the usual quasiconvexity condition of Morrey in two dimensions, which allows us to prove lower semicontinuity for functionals which are unbounded as the determinant vanishes. This notion, that we call…
The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical…
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal extension to the whole plane if its Schwarzian…
Given two metric spaces $\mathcal N \subseteq \mathcal M$ in inclusion and $0<p\leq 1$, we wish to determine the smallest constant $\mathfrak{t}_p (\mathcal N, \mathcal M)$ such that any Lipschitz map $f: \mathcal N \to Z$ into any…
We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible…
The present paper is devoted to the study of classes of mappings with non-bounded characteristic of quasiconformality. It is obtained a result on normal families of the open discrete mappings $f:D\rightarrow {\Bbb C}\setminus\{a, b\}$ of…
In this paper, we obtain new bounds for the Hausdorff dimension of planar elliptic measure via the application of quasiconformal mappings, with these bounds depending solely on the ellipticity constant of the matrix. In fact, in our case…
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…