61 research outputs found
Quasisymmetric parametrizations of two-dimensional metric spheres
We study metric spaces homeomorphic to the 2-sphere, and find conditions
under which they are quasisymmetrically homeomorphic to the standard 2-sphere.
As an application of our main theorem we show that an Ahlfors 2-regular,
linearly locally contractible metric 2-sphere is quasisymmetrically
homeomorphic to the standard 2-sphere, answering a question of Heinonen and
Semmes
Expanding Thurston Maps
We study the dynamics of Thurston maps under iteration. These are branched
covering maps of 2-spheres with a finite set of
postcritical points. We also assume that the maps are expanding in a suitable
sense. Every expanding Thurston map gives rise to a type of
fractal geometry on the underlying sphere . This geometry is represented
by a class of \emph{visual metrics} that are associated with the map.
Many dynamical properties of the map are encoded in the geometry of the
corresponding {\em visual sphere}, meaning equipped with a visual metric
. For example, we will see that an expanding Thurston map is
topologically conjugate to a rational map if and only if is
quasisymmetrically equivalent to the Riemann sphere . We
also obtain existence and uniqueness results for -invariant Jordan curves
containing the set . Furthermore, we
obtain several characterizations of Latt\`{e}s maps.Comment: 492 pages, 51 figure
The quasi-periods of the Weierstrass zeta-function
We study the ratio of the pseudo-periods of the Weierstrass
-function in dependence of the ratio of the
generators of the underlying rank-2 lattice. We will give an explicit geometric
description of the map . As a consequence, we obtain an
explanation of a theorem by Heins who showed that attains every value in
the Riemann sphere infinitely often. Our main result is implicit in the
classical literature, but it seems not to be very well known.
Essentially, this is an expository paper. We hope that it is easily
accessible and may serve as an introduction to these classical themes.Comment: 28 pages, 4 figures. Updated version. To appear in L'Enseignement
Math\'ematiqu
Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary
Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is
quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with
Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and
isometrically on hyperbolic 3-space.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper7.abs.htm
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