Topological equivalence for multiple saddle connections

Abstract

We study the topological equivalence between two vector fields defined in the neighborhood of the skeleton of a normal crossings divisor in an ambient space of dimension three. We deal with singularities obtained from local ones by ambient blowing-ups: we impose thus the nondegeneracy condition that they are all hyperbolic without certain algebraic resonances in the set of eigenvalues. Once we cut-out the attractors, we get the result if the corresponding graph has no cycles. The case of cycles is of another nature, as the Dulac Problem in dimension three.

Topological equivalence for chains of saddle-connections

“…The divisors D 1 and D 2 correspond to exceptional divisors under various blowing-up situations (see for instance [2,3,5,6]). In [4] the case of a single connection, without divisors, is studied with technics that are complementary to the ours.…”

“…In [4] the case of a single connection, without divisors, is studied with technics that are complementary to the ours. In [3] we consider multiple saddleconnections along the skeleton of a normal crossings divisor, under generic assumptions on the ratios of eigenvalues (see [7] for dimension two).…”