The infinitesimal 16th Hilbert problem in the quadratic caseGavrilov, Lubomir
doi: 10.1007/PL00005798pmid: N/A
Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and let
${X_H} = {H_y}\frac{\partial }{{\partial x}} - {H_x}\frac{\partial }{{\partial y}}$
be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of X
H
in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.
Thom polynomials, symmetries and incidences of singularitiesRimányi, Richárd
doi: 10.1007/s002220000113pmid: N/A
As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N
n
?P
n+k
for k>0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities.
Iteration of mapping classes and limits of hyperbolic 3-manifoldsBrock, Jeffrey F.
doi: 10.1007/PL00005799pmid: N/A
Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕ
i
X,Y)}
i=1
∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface D
ϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-D
ϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and D
ϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕ
si
X,Y)}
i=1
∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.