3,369 research outputs found
On Dynamics of Cubic Siegel Polynomials
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel
quadratic polynomials, we study the one-dimensional slice of the cubic
polynomials which have a fixed Siegel disk of rotation number theta, with theta
being a given irrational number of Brjuno type. Our main goal is to prove that
when theta is of bounded type, the boundary of the Siegel disk is a quasicircle
which contains one or both critical points of the cubic polynomial. We also
prove that the locus of all cubics with both critical points on the boundary of
their Siegel disk is a Jordan curve, which is in some sense parametrized by the
angle between the two critical points. A main tool in the bounded type case is
a related space of degree 5 Blaschke products which serve as models for our
cubics. Along the way, we prove several results about the connectedness locus
of these cubic polynomials.Comment: 58 pages. 20 PostScript figure
Conformal Fitness and Uniformization of Holomorphically Moving Disks
Let be a family of topological disks on the
Riemann sphere containing the origin 0 whose boundaries undergo a holomorphic
motion over the unit disk . We study the question of when there
exists a family of Riemann maps which depends
holomorphically on the parameter . We give five equivalent conditions which
provide analytic, dynamical and measure-theoretic characterizations for the
existence of the family , and explore the
consequences.Comment: 32 pages, 4 figure
On Margulis cusps of hyperbolic 4-manifolds
We study the geometry of the Margulis region associated with an irrational
screw translation acting on the 4-dimensional real hyperbolic space. This
is an invariant domain with the parabolic fixed point of on its boundary
which plays the role of an invariant horoball for a translation in dimensions
. The boundary of the Margulis region is described in terms of a
function which solely depends on the
rotation angle of . We obtain an
asymptotically universal upper bound for as for
arbitrary irrational , as well as lower bounds when is
Diophatine and the optimal bound when is of bounded type. We
investigate the implications of these results for the geometry of Margulis
cusps of hyperbolic 4-manifolds that correspond to irrational screw
translations acting on the universal cover. Among other things, we prove
bi-Lipschitz rigidity of these cusps.Comment: 34 pages, 6 figure
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