scholarly journals Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

2001 ◽  
Vol 83 (1) ◽  
pp. 289-312 ◽  
Author(s):  
Zoltán M. Balogh
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Quasiconformal Mappings

scholarly journals A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

2019 ◽  
Vol 199 (1) ◽  
pp. 147-186
Author(s):  
Tomasz Adamowicz ◽  
Katrin Fässler ◽  
Ben Warhurst
Keyword(s):  
Heisenberg Group ◽  
Quasiconformal Mappings ◽  
Distortion Theorem

scholarly journals Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

2013 ◽  
Vol 1 ◽  
pp. 232-254 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Jeremy T. Tyson ◽  
Kevin Wildrick
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Metric Space ◽  
Metric Spaces ◽  
Higher Dimension ◽  
Metric Measure Spaces ◽  
Regular Mapping ◽  
Sobolev Mappings ◽  
Measure Spaces ◽  
Left And Right

Abstract We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

Quasiconformal mappings on the Heisenberg group

1985 ◽  
Vol 80 (2) ◽  
pp. 309-338 ◽  
Author(s):  
A. Kor�nyi ◽  
H. M. Reimann
Keyword(s):  
Heisenberg Group ◽  
Quasiconformal Mappings

Uniform domains and quasiconformal mappings on the Heisenberg group

1995 ◽  
Vol 86 (1) ◽  
pp. 267-281 ◽  
Author(s):  
Luca Capogna ◽  
Puqi Tang
Keyword(s):  
Heisenberg Group ◽  
Quasiconformal Mappings ◽  
Uniform Domains

scholarly journals Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

2021 ◽  
pp. 1-34
Author(s):  
LAURENT DUFLOUX ◽  
VILLE SUOMALA
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Hyperbolic Plane ◽  
Strong Version ◽  
Empty Interior ◽  
Vertical Projection ◽  
Euclidean Metric ◽  
The One ◽  
One Point Compactification ◽  
Complex Hyperbolic Plane

Abstract We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\] , endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\] ) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\] . As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\] . We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\] , we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

scholarly journals Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group

2020 ◽  
Vol 8 (1) ◽  
pp. 15-35
Author(s):  
Fernando Román-García
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Positive Measure ◽  
Orthogonal Complement ◽  
Orthogonal Projections ◽  
Measurable Set

AbstractThis paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2n\B there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.

scholarly journals Stretching and Rotation of Planar Quasiconformal Mappings on a Line

2021 ◽  
Author(s):  
Olli Hirviniemi ◽  
István Prause ◽  
Eero Saksman
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Quasiconformal Mapping ◽  
Quasiconformal Mappings ◽  
Optimal Estimate ◽  
Holomorphic Motions

AbstractIn this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline {B}(1/(1-k^{4}),k^{2}/(1-k^{4}))$ B ¯ ( 1 / ( 1 − k 4 ) , k 2 / ( 1 − k 4 ) ) . This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension 1. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a 1-dimensional subset of a line under a quasiconformal mapping.

scholarly journals Foundations for the Theory of Quasiconformal Mappings on the Heisenberg Group

1995 ◽  
Vol 111 (1) ◽  
pp. 1-87 ◽  
Author(s):  
A. Koranyi ◽  
H.M. Reimann
Keyword(s):  
Heisenberg Group ◽  
Quasiconformal Mappings
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