Infinitesimal Hartman-Grobman Theorem in Dimension Three

In this paper we give the main ideas to show that a real analytic vector field in R3 with a singular point at the origin is locally topologically equivalent to its principal part defined through Newton polyhedra under non-degeneracy conditions.


INTRODUCTION
Let ξ = a(x, y, z) @ + b(x, y, z) @ + c(x, y, z) @ @x @y @z be a real analytic vector field defined in a neighbor hood of the origin of R 3 and assume that the origin is an equilibrium point of ξ.For hyperbolic singularities, the Hartman-Grobman theorem establishes that ξ is locally topologically equivalent to its linear part.If the linear part is null, it is a natural question to ask for a representative of the topological type of ξ around the origin.
In dimension two this problem was solved for C ∞ vector fields by Brunella and Miari: Theorem.(Brunella and Miari 1990) Let ξ be a plane C ∞ vector field with ξ (0) = 0 and non-degenerate principal part Pξ such that 0 is an isolated singularity of Pξ; then ξ is locally topologically equivalent to Pξ modulo center-focus.
The proof of this result is based on the construction of a morphism obtained from the Newton polygon of ξ.This morphism is a sequence of blow-ups centered at singular points.Under nondegeneracy conditions it desingularises both ξ and its principal part.Moreover, given that there is no return, they found a topological equivalence around the exceptional divisor between the transformed vector fields ξ e and Pξ f .By projection they get the desired homeomorphism.
In dimension three the results of M.I.Camacho (Camacho 1985) and BonckaertDumortierVan Strien (Bonckaert et al. 1989) show that, under non degeneracy conditions, the first nonvanishing jet j k ξ(0) of ξ at the origin determines the topological type of ξ.Their particular case co rresponds to a Newton polyhedron with a unique compact face per pendicular to the vector (1, 1, 1) 2 R 3 .In this situation, the principal part of ξ is the homogeneous vector field j k ξ(0) and the desingularisation morphism consists of just one blowup centered at the origin.
In this paper we give an idea of the proof of the following result: Theorem 1.Let ξ be a three dimensional real analytic germ of vector field with absolutely isolated This result can be considered as an infinitesimal version of the classical Hartman-Grobman theorem in dimension three.
Absolutely isolated singularities of vector fields were introduced for the complex case by Camacho-CanoSad in (Camacho et al. 1989).The definition is similar in the real case: the singularity is isolated and, after a finite sequence of blowups with center at singular points, we get isolated singularities and the exceptional divisor is invariant.In (Camacho et al. 1989) it is also proved that we obtain a reduction of singularities of the vector field after a finite number of blow-ups.This result also holds in the real case.
A germ of vector field with absolutely isolated singularity has a Newton polyhedron of barycentric type up to change of coordinates.This type of polyhedra gives a finite sequence of combinatorial blowups (centered at the origin of the charts).Fixed such a polyhedron N, we say that the principal part Pξ of ξ given by N is non-degenerate if the associate sequence of blow-ups π N is a desingularisation of Morse-Smale type of Pξ.In this situation, we have that π N is also a desingularisation of MorseSmale type of ξ.
The definition of MorseSmale type desin gularisation is detailed later on.This non-degeneracy condition is a three-dimensional version of the classical MorseSmale ones and generalize the conditions imposed in (Camacho 1985) and (Bonck aert et al. 1989).We also ask for an infinitesimal nonreturn condition over Pξ : given any small transversal section Σ to D and Pξ f , there is a neighborhood of D such that each orbit of Pξ f cuts Σ at most once.This is the analogous to the center-focus exclusion in (Brunella and Miari 1990).
Finally, we use the study done by Alonso-Gonzalez et al. 2006, 2008 about the topological classification of vector fields whose reduction of singularities is of MorseSmale type, to get the desired topological equivalence.

BARYCENTRIC TYPE POLYHEDRA
In dimension two the Newton polygon of a vector field ξ determines a sequence of blow-ups with center at points.Moreover, if the singularity is of toric type, this morphism is a reduction of singularities of ξ up to change of coordinates (Camacho and Cano 1999).In dimension n ≥ 3 this result does not work in general but barycentric type polyhedra naturally gives a sequence of combinatorial blowups.Moreover, vector fields with absolutely isolated singularity have a Newton polyhedron of barycentric type.
Given a vector field ξ in R 3 with ξ(0) = 0, let us consider the associated Newton polyhedron N = N(ξ) in fixed coordinates.Taking normal vectors to each face of N we get a set of cones whose union is a fan ∆ N .The standard fan ∆ st has the first octant R 3 ≥ 0 as a unique cone.By construction, ∆ N is a subdivision of ∆ st .Denote this by ∆ N >> ∆ st .
We say that a polyhedron N is barycentric if ∆ N is barycentric: it is obtained from successive barycentric subdivisions of ∆ st .Moreover, we say that N is of barycentric type if there is a barycentric fan B with B >> ∆ N .Clearly a barycentric type fan ∆ is not barycentric in general but there is a "minimal" barycentric fan B ∆ refining it (see Figure 1).
From toric geometry theory we know that the first barycentric subdivision of ∆ st into the cones b 1 , b 2 , b 3 corresponds to the blow-up of R 3 with center at the origin (see Oda 1985).As a consequence, if ∆ N is of barycentric type, we can consider a sequence of barycentric fans

NON-DEGENERACY CONDITIONS
Let N be a fix barycentric type polyhedron and π N the associated sequence of blow-ups.Recall that the Hartman-Grobman theorem holds for vector fields ξ having a hyperbolic singular point.Brunella and Miari also worked under some nondegeneracy conditions satisfied by Pξ (see Brunella and Miari 1990).In dimension three, in case the principal part is homogeneous, M.I.Camacho in (Camacho 1985) and BonckaertDumortierVan Strien in (Bonckaert et al. 1989), assumed the classical MorseSmale conditions over ξ e | D (only hyperbolic singular points and no two-dimensional saddle-connections) so that π N (only one blow-up) desingularises Pξ and ξ.
In our situation more than one blow-up is involved in π N and the exceptional divisor D has more than one irreducible component.The dynamics around D is much more complicated and additional conditions have to be imposed: we assume that π N is a desingularisation of Pξ of Morse-Smale type.Definition 1.We say that π N is a desingularisation of Pξ of Morse-Smale type if three conditions are satisfied: (1) All the singular points on the exceptional divisor are hyperbolic.
(2) Two dimensional saddle-connections are not allowed out of the skeleton of D (the intersection of divisor irreducible components).
Let us explain the two last conditions that correspond to concepts already introduced in (Camacho 1985) and (AlonsoGonzalez et al. 2008): Two-dimensional saddle-connections.Recall that a two-dimensional saddle-connection appears when two saddles are connected along their unstable-stable varieties.The second condition means that given a component D i of the exceptional divisor, there are no two-dimensional saddle-connections of ξ e | Di along unstable-stable varieties contained exclusively in the component D i .That is, we only allow the existence of two-dimensional saddle-connections of ξ e | D along the skeleton of D. This last situation is rigid in the sense that it is preserved under the usual deformations of the vector field (based on Melnikov's integral) addressed to destroy interior saddle-connections.For details see (Camacho 1992).
Infinitesimal saddle-connections.The third condition involves three dimensional saddles.To explain it we need to recall some concepts.Suppose that the linear part of a vector field ξ with a saddle at p is
If we start with the intrinsic weight α of a corner p, by means of the previous rule of weights transition, the value α transits through connected saddles producing an associated weight at each step.Two situations are possible: (1) The process does not stop at any saddle.
(2) There is a saddle q (on the skeleton) where the transition stops, i.e. the obtained value by transition coincides with the intrinsic weight of q.In this case we say that p and q determine an infinitesimal saddleconnection (for details, see AlonsoGonzalez et al. 2008).
The principal part Pξ of a vector field ξ given by a fixed polyhedron N has a finite number of coefficients.Hence the set P N of associated principal parts to N is isomorphic to a ndimensional affine space R n .Given N, there is a "generic" set G ⊂ R n of non-degenerate principal parts.On the other hand, under the non-degeneracy condition, the restrictions to the exceptional divisor of ξ e and Pξ f coincide.Hence we can conclude the following result: Theorem 3. Let N be a barycentric type polyhedron in R 3 and P N the set of associated principal parts.There is a nonempty set G ⊂ P N such that for any vector field ξ with ξ(0) = 0 whose principal part Pξ belongs to G, the morphism π N is a reduction of singularities of Morse-Smale type of ξ and Pξ.

CONSTRUCTION OF THE TOPOLOGICAL EQUIVALENCE
Given a vector field ξ with ξ(0) = 0 and Pξ 2 G, the last step to generalize the BrunellaMiari result to a three dimensional space, is the construction of the topological equivalence between ξ and Pξ around the singular point.We use the process described in the papers AlonsoGonzalez et al. 2006AlonsoGonzalez et al. , 2008 to determine the π N -topological type of ξ (topological type after desingularization).There the reader can find a complete topological classification in the class of three-dimensional real analytic vector fields whose reduction of singularities is of Morse Smale type without infinitesimal return.Let us recall the principal ideas.
Given ξ and ξ' vector fields having the same reduction of singularities.Supposse that it is of Morse Smale type.The main difficulty in the construction of the topological equivalence after desingularisation is the appearance of saddle connections along the skeleton of D.Even in the case of just two saddles connected along their one dimensional invariant variety, three topological types could appear (see AlonsoGonzalez 2003).The key is that if the weights of ξ e and ξ' e obtained by the previous transition rule at each singular point are well ordered and, infinitesimal As a consequence, the π N -topological type of a vector field ξ with reduction of singularities of MorseSmale type without infinitesimal return depends only on the eigenvalues of ξ e at the singularities and on the topological type of the restriction ξ e | Di to the irreducible components D i of the exceptional divisor.
In our case, if Pξ is non-degenerate, the eigenvalues of ξ e and Pξ f coincide at each singular point.Hence the weights also coincide.Moreover, we have that Pξ f | Di = ξ e | Di .The no infinitesimal return condition is also determined by Pξ f .Summing up all the previous ideas, we have the following result: Theorem 4. Let N be a barycentric type polyhedron in R 3 and P N the set of associated principal parts.Then, there is a nonempty set G ⊂ P N of genericity such that any vector field ξ with ξ(0) = 0 whose principal part Pξ belongs to G is topologically equivalent to Pξ around the origin modulo infinitesimal return.
Given that absolutely isolated singularity implies barycentric type polyhedron and taking G as the set of non-degenerate principal parts, Theorem 1 is a consequence of this last result.

ACKNOWLEDGMENTS
This research was partially supported by the Spanish Government (MTM2010 15471 (subpro grama MTM)).
and then a sequence π N of blow-ups centered at points.Theorem 2. Let ξ be an analytic three dimensional real vector field.If the origin is an absolutely isolated singularity of ξ, then the associated Newton polyhedron N = N(ξ) is of barycentric type.Note that if N = R 3 ≥ 0 we are done.Otherwise, as the origin is an isolated singularity, the union of the compact faces of N cuts the coordinate INFINITESIMAL HARTMAN-GROBMAN THEOREM IN DIMENSION THREE axes in three points: (a, 0, 0), (0, b, 0), (0, 0, c).This fact repeats after blowing-up given that the singularity is absolutely isolated.Let us denote h = h(N) = a + b + c.Now, if ξ i e is the strict transform of ξ after a blow-up centered at the origin of the i-chart and N i is its Newton polyhedron, we have that h(N i ) < h(N).Besides, the fan ∆ i associated to ξ i e is isomorphic to the fan ∆ N ∩ b i .We conclude taking into account that ∆ N =[ 3 i=1 ∆ N ∩ b i and working by induction over h.

Figure 1 -
Figure 1 -A barycentric fan and a barycentric type fan.