Topological equivalence for chains of saddle-connections

Abstract

In this paper we give a complete topological classification for multiple saddle-connections of a real analytic vector field along an axis G in an ambient space of dimension three, under the assumption that G is the intersection of two invariant surfaces D 1 and D 2 : r

Topological equivalence of vector fields after blow-up

“…Suppose that the linear part of an H -vector field ξ whit a saddle point at Q is [1]) and we say that it is attached to the x-axis. Now we describe a process of weights transition ρ → ρ introduced in [2,3].…”

“…This saddle has to be necessarily the repeller R j of a component D j (the arriving value from A i coincides with the intrinsic weight of R j ). In this case, we say that there is a saddle-connection S(D i , D j ) (see [1]). Recall that, as divisor components, the transformed of H could also take part in a saddle-connection.…”

“…In order to guarantee the compatibility of the topological equivalences obtained at the induction step, we look for those preserving a pasting homeomorphism h between transversal sections to one of the axis. First, we have to give this h in a convenient way: it has to preserve the set of (Φ, ξ )-weights attached to the corresponding axis (see [1][2][3]). Note also that the conditions on the statement imply that the origin is a singularity of the same type and the same orientation both for ξ and ξ .…”

“…If n = 0, the difficulty is to find a topological equivalence preserving an h between transversal sections to the one-dimensional invariant variety of a saddle. As explained in [1], it is enough to take this h preserving the respective intrinsic weights α and α of the origin. This means that if π α : [0, 1] 2 → D ++ is the weighted blow-up of the first quadrant: π α (r, t) = (r cos πt/2, r α sin πt/2), then h lifts by π α , π α .…”

Invariants topologiques des singularités de champs réels

“…. , Γ k , P k } où les Γ i sont des lignes pseudo-squelettiques qui ont comme bouts P i−1 et P i (lorsqu'ils existent) et P i ∈ Λ. Dans le second cas, nous dirons que les selles P 0 et P k sont infinitésimalement connectées (voir [1] pour un premier exemple de l'influence de ces connexions).…”