SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
FELIPE CANO
Contents
0. Introducción
1. Trayectorias de campos de vectores analı́ticos reales
1.1. El espacio ambiente
1.2. El flujo de un campo de vectores
1.3. Rectificación, puntos de equilibrio
1.4. Poincaré-Bendixson
1.5. Cambio de espacio ambiente, explosiones
1.6. Tangents and iterated tangents
1.7. Trajectories without iterated tangents
1.8. Oscillation and iterated tangents. Spiraling axes
2. Hardy Fields, valuations and non oscillating trajectories
2.1. Valuations
2.2. The center of a valuation
2.3. Hardy Fields
2.4. Non oscillating trajectories and Hardy fields
2.5. Galois packages
2.6. Dancing trajectories and separated packages
3. Valuations in dimension two. Bamboos and desingularization of a
foliation
4. Applications in dimension three. Dichotomy separation-linking for
Galois packages. Local uniformization. Global results of Panazzolo.
5. Codimension one foliations and vector fields along a formal curve.
Examples of F. Sanz y F. Sancho
6. Ends of leaves and local Brunella’s alternative
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(Initial draft, surely completely wrong ... )
0. Introducción
En este curso no pretendemos desarrollar una teorı́a sistemática. Ya hay otros
cursos previstos para ello en el contexto de las foliaciones y también en las valoraciones. Abordaremos una serie de temas con un “aroma” valorativo en el estudio
de sistemas dinámicos holomorfos o analı́ticos reales, sobre los cuales el autor ha
contribuido más o menos directamente.
Date: Diciembre 2014.
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Comenzaremos con un estudio muy geométrico de trayectorias de campos de
vectores en dimensión dos y tres por medio de explosiones del espacio. El caso no
oscilante enlaza directamente con las valoraciones asociadas a cuerpos de Hardy.
Presentaremos, de cara a posibles resultados de desingularización, el planteamiento
general de la uniformización local de Zariski y resultados de campos de vectores en
esa dirección, resultados que solamente existen en dimensión tres ambiente, salvo
para las singularidades absolutamente aisladas. Terminaremos presentando algunas
situaciones recientemente estudiadas en términos de finales de hojas y su similitud
con las valoraciones y las trayectorias no oscilantes.
1. Trayectorias de campos de vectores analı́ticos reales
1.1. El espacio ambiente. La geometrı́a no tiene mucho sentido si no se desarrolla en un espacio. Puede estudiar propiedades de figuras de manera intrı́nseca,
pero siempre tenemos presente la existencia de un espacio ambiente más o menos
implı́cita. Mucho más evidente es la necesidad de este ambiente en mecánica o
en el estudio de los sistemas dinámicos. El espacio ambiente más natural es el
relacionado con los polinomios, la topologı́a de Zariski del espacio afı́n, y de manera
global el espacio proyectivo, sin olvidarnos de la distancia euclı́dea. La mayorı́a de
las ecuaciones diferenciales que describen fenómenos interesantes tienen coeficientes
polinómicos, aun cuando sus soluciones deban describirse en otras categorı́as o sean
literalmente imposibles de describir con detalle. Además los polinomios permiten
usar toda la fuerza de la geometrı́a algebraica. Las funciones analı́ticas, descritas
por series convergentes guardan todavı́a algunas de esas propiedades de rigidez
deseables para un espacio ambiente.
En estas notas tomaremos como espacios ambiente o bien variedades algebraicas
(preferentemente proyectivas) o bien variedades analı́ticas, sobre el cuerpo de los
números reales o sobre el cuerpo de los múmeros complejos. Supondremos el lector
familiarizado con la descripción de estos espacios en términos de cartas locales y con
nociones de su descripción como espacios anillados, sobre todo relativas al proceso
de germificación de secciones de haces.
1.2. El flujo de un campo de vectores. Sea M una variedad analı́tica real y
consideremos un campo de vectores analı́tico ξ definido sobre M . En coordenadas
locales x1 , x2 , . . . , xn centradas en un punto p ∈ M , el campo de escribe
ξ = a1
∂
∂
∂
+ a2
+ · · · + an
∂x1
∂x2
∂xn
donde a1 , a2 , . . . , an son funciones analı́ticas definidas en un entorno de p. Decimos
que una curva parametrizada
γ : (−ϵ, ϵ) → M
es una curva integral de ξ centrada en p si γ(0) = p para todo t ∈ (−ϵ, ϵ)se tiene
que
d
γ ′ (t) = dt ( ) = ξ(γ(t)).
dt
Es decir, el vector tangente, o vector “velocidad” γ ′ (t) de γ en t es el vector seleccionado por el campo de vectores ξ en el punto γ(t).
SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
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En términos de ecuaciones diferenciales, escribiendo ai = ai (x1 , x2 , . . . , xn ) las
condiciones anteriores equivalen a decir que ai (0, 0, . . . , 0) = 0 y que
γi′ (t) = ai (γ1 (t), γ2 (t), . . . , γn (t))
para todo i = 1, 2, . . . , n. Este es un sistema (autónomo) de ecuaciones diferenciales
ordinarias con condiciones iniciales, un “problema de Cauchy”. Sabemos que tiene
solución γp y que esta es única en el sentido de que dos soluciones coinciden en su
domino común de definición. Más precisamente, existe una función analı́tica
Fξ : U × (−ϵ, ϵ) → M
donde U ⊂ M es un entorno abierto de p en M tal que para todo q ∈ U se tiene
γq (t) = Fξ (q, t). Diremos que Fξ es el flujo de ξ. Evidentemente corresponde a la
“dinámica” que permite mover cada punto q ∈ U un tiempo t ∈ (−ϵ, ϵ):
q 7→ γq (t).
La relación fácilmente comprobable
γγq (t) (s) = γq (t + s)
(para t, s suficientemente pequeños), justifica el nombre de grupo uniparamétrico
local para el conjunto de los flujos Fξ .
Se puede ver que si M es compacto, el flujo está definido para todo t ∈ R y
entonces tenemos un flujo global (el campo se llama completo, en el caso complejo
tienen mucho interés, pero no entraremo en ello). Las trayectorias del flujo son las
imágenes de las curvas integrales, que podemos considerar orientadas o no (aunque
a veces les supondremos también parametrización). El mismo teorema de existencia y unicidad ya citado nos permite ver que el espacio es una unión disjunta de
trayectorias.
Comprender aspectos de la distribución en el espacio de estas trayectorias en
uno de los objetivos principales del estudio de los sistemas dinámicos.
1.3. Rectificación, puntos de equilibrio. Supongamos que ξ(p) ̸= 0, es decir,
el campo de vectores ξ selecciona en el punto p un vector tangente ξ(p) ∈ Tp M que
es no nulo. El clásico teorema de rectificación determina que existe un sistema de
coordenadas locales x1 , x2 , . . . , xn , centrado en p tal que en su abierto de definición
U (que puede tomarse “cuadrado”) se tiene que
∂
.
∂x1
Dicho de otro modo, leı́do en dichas coordenadas el flujo actúa como un desplazamiento horizontal a velocidad uno
ξ|U =
((λ1 , λ2 , . . . , λn ), t) 7→ (λ1 + t, λ2 , . . . , λn ).
En particular, dentro del abierto U las trayectorias están dadas por
x2 = λ2 , x3 = λ3 , . . . , xn = λn .
El abierto U se llama caja de flujo.
Ası́ pues, desde el punto local, la dinámica en un punto con ξ(p) ̸= 0 está completamente descrita. Otra cosa es la presencia de holonomı́a debida a la prolongación
de trayectorias fuera de la caja de flujo (pasando por otras cajas de flujo) y que
vuelven a la caja original; la holonomı́a, que depende de la estructura global del
espacio ası́ como de la de las trayectorias (esto último en el caso complejo o en
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dimensiones superiores) es una herramienta básica en el estudio de los sistemas
dinámicos, útil incluso a nivel local como veremos.
Sin embargo, la situación es mucho más complicada cuando ξ(p) = 0. En este
caso la curva integral es
γp (t) = p
para todo t ∈ R. Es decir, no hay movimiento. Por eso se dice que p es un punto
de equilibrio o un punto singular. Ciertamente, la dinámica en el punto p está fija,
pero qué podemos decir de la dinámica en un entrono de p. No podemos situar p en
una caja de flujo, ası́ que en los puntos no singulares próximos a p tendremos cajas
de flujo “cada vez más pequeñas” y podremos observar fenómenos de holonomı́a a
nivel local o incluso situaciones más complejas.
Si una curva integral γ(t) se aproxima a un punto de equilibrio, lo hace en tiempo
infinito, es decir asintóticamente. Más generalmente, se define el conjunto ω-lı́mite
ω(γ) de γ como la intersección de las adherencias de γ(a, +∞) cuando a → ∞.
Uno de los aspectos que desarrollaremos en estas notas es el estudio de trayectorias cuyo conjunto ω-lı́mite se reduce a un punto. Realizaremos un estudio tanto
una por una como en paquetes de Galois, como señalaremos más adelante.
1.4. Poincaré-Bendixson. El conjunto ω-lı́mite no tiene porqué reducirse a un
punto. El resultado más clásico sobre el aspecto que puede ofrecer un conjunto
ω-lı́mite en un ambiente de dimensión dos es el teorema de Poincaré-Bendixson
que describe estos posibles conjuntos sobre la superficie esférica. No lo vamos a
enunciar completamente aquı́, pero llamaremos la atención sobre la posibilidad
de que el conjunto ω-lı́mite sea una órbita periódica, es decir, una curva integral
que “vuelve sobre sı́ misma” cumpliendo γ(t + t0 ) = γ(t). En este caso la órbita
periódica es homeomorfa a la circunferencia y la trayectoria que se acumula sobre
ella lo hace solo de uno de los lados señalados por el teorema de la curva de Jordan.
1.5. Cambio de espacio ambiente, explosiones. Para entender cómo se organiza la dinámica alrededor de un punto singular, es útil poner una “lupa infinitesimal” en el punto. Usualmente esto se hace mediante una o sucesivas explosiones del
espacio ambiente. Hay dos tipos de explosiones, la correspondiente a coordenadas
polares en el caso real, llamada también explosión real orientada y la explosión no
orientada, válida en todos los contextos de la geometrı́a algebraica, que substituye
el punto por el espacio proyectivo formado por las direcciones de aproximación al
mismo.
La explosión orientada con borde del origen de Rn es la aplicación
σ : Sn−1 × R≥0 → Rn ;
(x, t) 7→ tx.
La imagen inversa de una semirecta determina un único punto de σ −1 (0) y fuera
de σ −1 (0) es una aplicación biyectiva. Esta misma aplicación vista como
σ : Sn−1 × R → Rn
es la explosión orientada doble (sin borde). Se dice doble porque fuera del origen
es dos a uno, por ejemplo, la imagen inversa de una recta es la misma que la de
cualquiera de las dos semirrectas que la componen.
Haciendo una identificación de una carta local en p ∈ M con Rn e identificando
por U \ {p}, la anterior explosión define una aplicación σ : M̃ → M , explosión real
de M con centro en p, que podemos considerar con borde o sin borde. En ambos
casos la situación se puede iterar y podemos hablar de explosiones sucesivas.
SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
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Demos un ejemplo para n = 2 de cómo se puede expresar σ en ciertas cartas.
Podemos cubrir S 1 × R por cuatro cartas U1+ , U1− , U2+ , U2− , donde
U1− = {eis ; π/2 < s < 3π/2} × R
U1+ = {(eis ; −π/2 < s < π/2)} × R,
U2+
U2− = {eis ; π < s < 2π} × R
= {(eis ; 0 < s < π)} × R,
y las aplicaciones coordenadas están dadas respectivamente por
± ±
is
ϕ±
1 : (e , t) 7→ (x1 , y1 ) = (t cos s, tan s)
± ±
is
ϕ±
2 : (e , t) 7→ (x1 , y1 ) = (1/ tan s, t sin s)
En estas coordenadas, la explosión se expresa
x =
y =
x±
1
± ±
x1 y1
±
x±
2 y2
±
y2
=
=
Son las mismas expresiones algebraicas que definirán la explosión no orientada, pero
las cartas no son las mismas, en la explosión no orientada se usarán solo dos cartas.
Ahora podemos observar en estas cartas cómo se transforma un campo de vectores con centro en un punto de equilibrio, sin más que observar que si x = x′ , y =
x′ y ′ , debemos considerar que
x
∂
∂
∂
+y
= x′ ′ ;
∂x
∂y
∂x
y
∂
∂
= y′ ′ .
∂y
∂y
Ası́, si tenemos
ξ = a(x, y)
∂
∂
+ b(x, y)
∂x
∂y
el campo transformado se escribe
(
)
a(x′ , x′ y ′ )
b(x′ , x′ y ′ ) ∂
′ ∂
′ ∂
ξ=
x
−
y
+
.
x′
∂x′
∂y ′
x′
∂y ′
Nótese que x′ divide a(x′ , x′ y ′ ) y también b(x′ , x′ y ′ ) dado que a(0, 0) = b(0, 0) = 0.
1.6. Tangents and iterated tangents. Take a trajectory γ of an analytic vector
field
∂
∂
∂
ξ = a1
+ a2
+ · · · + an
∂x1
∂x2
∂xn
defined in a neighborhood of the origin 0 ∈ Rn . We assume that the origin is the
only ω-limit point of γ, that is
lim γ(t) = {0}.
t→∞
We can ask if γ arrives to the origin in a privilegied oriented direction, that is, we
wonder if there is a limit of the secants
γ(t)/∥γ(t)∥ ∈ Sn−1
as t goes to +∞. Such a limit shoudl be called the oriented tangent of γ, or simply
the tangent of γ. The existence of tangent is not always assured. For instance, in
dimension two we can consider examples as
ξ=x
∂
∂
−y ;
∂y
∂x
ξ = (x − y)
∂
∂
− (x + y)
∂y
∂x
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where we have a center of a spiraling situation, and evidently there are no tangent
for the trajectories. In dimension three we can also produce examples with many
trajectories without tangent as
ξ = (x − y)
∂
∂
∂
− (x + y)
−z .
∂y
∂x
∂z
Note that ξ in this example is tangent to the conic f = x2 + y 2 − z 2 = 0, since we
have η(ξ) = f g, where
η = d(x2 + y 2 − z 2 ) = 2xdx + 2ydy − 2zdz.
The polar blowing-up σ : Sn−1 × R≥0 → Rn−1 revels the existence of tangents
as follows. Take γ̃ = σ −1 ◦ γ the lifting of γ (it exists because γ never arrives to the
origin). Then γ has a tangent if and only if the ω-limit set ω(γ̃) is only one point
p′ ∈ Sn−1 × {0}. This point p′ may be called the tangent γ.
If we have a tangent, the procedure may be iterated. We blow-up, we consider
the tranformed trajectory (and vector field) at the tangent p′ and we perform a
new blowing-up with center p′ . In this way we can produce iterated tangents or
not. Some trajectories will have the property of the iterated tangents. They are
the trajectories we are mainly interested in this notes.
1.7. Trajectories without iterated tangents. There is an open question related
with an infinitesimal version of Poincaré-Bendixson Theorem. Suppose we have a
trajectory γ with the origin as only ω-limit point. Let us do the blow-up
σ : Sn−1 × R≥0 → Rn−1
and consider the ω-limit set ω(γ̃) of the lifted trajectory γ̃. It is a connected and
compact subset of Sn−1 , the set of accumulation of secants. The question is: what
is the structure of this set?. We have a complete answer in dimension two: the
accumulation set is just one point or the whole S1 . Let us give some lines on the
proof of this.
Let us do the blow-up σ : S1 × R≥0 → R2 of the origin of R2 . Suppose that the
set ω(γ̃) of accumulation of secants is not a single point, nor the whole S1 . The only
possibility is that it is a compact interval [α, β] of S1 . Now, we consider two points
(a, 0), (b, 0) ∈ S1 × {0} belonging with a, b ∈ (α, β) not singular for the transformed
˜ Consider the lines
vector field ξ.
La = {a} × R, Lb = {b} × R.
The vector field ξ˜ is transversal to La , Lb (otherwise the problem is easy) and it
determines an orientation near (a, 0), (b, 0). It must be the same orientation, since
otherwise γ̃ cannot accumulate at the points in [α, a) and (b, β]. But in order to
return to the points in (a, b), the trajectory must go to the exterior of [α, β] and
thus it accumulates at all of S1 .
The above argument is a key of the proof of Poincaré-Bendixson Theorem and it
is also extensively used in our presentation of the oscilation and spiraling properties.
There are other methods using for instance the Theory of Khovanskii and the
concept of Pfaffian leaf.
In dimension three we have some results for particular vector fields, but a general
statement is missing.
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1.8. Oscillation and iterated tangents. Spiraling axes. We have already
done an argument observing that a trajectory cuts a line infinitely many times.
Take a trajectory γ of an analytic vector field ξ over Rn such that ω(γ) = {0}.
We say that γ is non-oscillating with respect to analytic sets if and only if for any
analytic hypersurface S = (f = 0) there is a time t0 such that one of the following
properties holds:
(1) γ(t) ∈
/ S for any t ≥ t0 .
(2) γ(t) ∈ S for any t ≥ t0 .
If in addition the second situation never arrives, we say that γ is transcendental.
The first observation is that if γ is non oscillating, then it has the property of
the iterated tangents. If it is not so, after finitely many blowing-ups, we find two
points of accumulation. We can put an analytic “wall” between them that must be
crossed infinitely many times by the trajectory.
In dimension two, we also have that the property of the iterated tangents implies
non oscillation. It is an argument similar to the previous one. We do a sketch of it.
Up to reduction of singularities of curves, we may assume that we have oscillation
with respect to a half line L. For t >> 0 the trajectory cuts L always with the
same orientation, otherwise we find infinitely many points of tangency between L
and ξ by the classical theorem of the mean value and, by analyticity, all the half
curve L must be invariant, absurd. Now, we blow-up and we find as above that the
omega limit set of the trajectory must be the whole exceptional divisor σ −1 (0).
Nevertheless, it is possible to find examples in dimension three of oscillation in
presence of the property of the iterated tangents. For instance
ξ = (x − y)
∂
∂
∂
− (x + y)
− z2 .
∂y
∂x
∂z
The trajectories are oscillating with respect to any plane containing the z-axis.
All that trajectories have a flat contact with the z-axis and, after blow-up, their
iterated tangents follow the strict transform of the z-axis.
The trajectories that are oscillating and have the iterated tangents at the same
time give rise to a new geometrical object that organizes the dynamics: the axes
of spiraling. We say that a germ of semi-analytic curve Γ at the origin of R3 is an
axis of spiraling for a trajectory γ with ω(γ) = {0} when the following holds: for
any germ of analytic surface S ⊂ (R3 , 0) we have that γ is oscillating with respect
to S if and only if Γ ⊂ S.
The axes of spiraling, if they exists, they are unique, moreover, γ is necessarily
oscillating and it has the property of iterated tangents, more precisely, the iterated
tangents of γ are given by the successive strict transforms of Γ. Moreover, up to
blow-up enough times, we can assume that the axis is the z-axis and that γ goes
to the origin “turning infinitely around” Γ.
The main result we have in this direction is the following one
Let γ be a trajectory of an analytic germ of vector field ξ at the
origin 0 ∈ R3 such that ω(γ) = {0}. If γ is oscillating and has the
property of the iterated tangents, then there is an axis of spiraling
Γ for γ.
How we obtain the axis of spiraling? First, we consider a surface S such that γ
oscillates with respect to S. Taking points between two points of oscillation of γ
with S, we find analytic curves Γ1 , Γ2 , . . . , Γn of tangency between ξ and S. By
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FELIPE CANO
performing blowing-ups following the iterated tangents of γ, we follow one of that
curves, say Γ and this after any number of blowing-ups. Now, to find that Γ is an
axis of spiraling we need to do a stratification of S with respect to Γ and to observe
that the only possible oscillation (Γ is the only curve of tangency) is by spiraling
around Γ: hence we have oscillation with respect to any surface containing Γ.
2. Hardy Fields, valuations and non oscillating trajectories
2.1. Valuations. Let K be a field and k ⊂ K a subfield. The typical example we
have in mind is k = C or k = R and K the field of rational functions of a projective
algebraic variety over k, that is
K = k(t1 , t2 , . . . , tn )
is a finitely generated field extension of k. As it has been popularized around me
by M. Spivakovsky, a k-ring of valuation for K is a local subring (R, MR ⊂ K such
that k ⊂ R and for any ϕ ∈ K we have that ϕ ∈ R or 1/ϕ ∈ R.
We can do an equivalence relation between the elements of K \ {0} by saying
that ϕ ∼ ψ if and only if ϕ/ψ ∈ R \ MR , that is ϕ/ψ is a unit in R. The canonical
map
ν : K \ {0} 7→ Γ = K \ {0}/ ∼
is a morphism of abelian groups for the law in Γ defined by posing
[ϕ] + [ψ] = [ϕψ].
Moreover, the group Γ is an ordered group if we define
Γ>0 = {[ϕ]; ϕ ∈ MR }.
Note that R = {ϕ ∈ K; ν(ϕ) ≥ 0} and MR = {ϕ ∈ K; ν(ϕ) > 0}. We have of
course the classical axioms of a valuation
(1) ν(ϕψ) = ν(ϕ) + ν(ψ).
(2) ν(ϕ+ψ) ≥ min{ν(ϕ), ν(ψ)} and ν(ϕ+ψ) = min{ν(ϕ), ν(ψ)} if ν(ϕ) = ν(ψ).
These properties recall very closely the classical comparison of real functions going
to the infinity. This is not a coincidence as we shall see.
2.2. The center of a valuation. Suppose now that K is the field of rational
functions k(V ) of an algebraic variety V over k and let k ⊂ R ⊂ K be a valuation
ring. Any projective variety V ⊂ Pnk such that k(V ) = K is called a projective
model of K. Recall that if Y ⊂ V is an irreducible closed subvariety the local ring
OV,Y is defined by
OV,Y = {ϕ ∈ K; Y ̸⊂ poles of ϕ}.
That is, the local ring is the ring of rational functions defined in a generic point of
Y , hence in a Zariski open set of Y . We say that Y is a center of R in the projective
model V if OV,Y ⊂ R and the maximal ideal MY of OV,Y satisfies
MY = MR ∩ OV,Y .
The center, if it exists is unique. Let us see this. Suppose we have two centers Y1 and
Y2 . It is known that, up to change the order, there is a rational function ϕ defined
both in a generic point of Y1 and a generic point of Y2 , that is ϕ ∈ OV,Y1 ∩ OV,Y2 ,
such that Y1 ⊂ Z(ϕ) and Y2 ̸⊂ Z(ϕ), where Z(ϕ) is the set of zeroes of ϕ. In other
words
ϕ ∈ ϕ ∈ OV,Y1 ∩ OV,Y2 ; ϕ ∈ MY1 \ MY2 .
SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
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This is a contradiction, since we have ϕ ∈ MR and ϕ ̸∈ MR at the same time.
Let us show the existence of a center. Let H = k[x0 , x1 , . . . , xn ] be the ring of
homogeneous coordinates of V . In the affine chart xi ̸= 0, the affine variety Vi has
the ring of regular functions given by
Ai = k[x0 /xi , x1 /xi , . . . , xn /xi ].
Up to reordering, assume that A0 has the maximal number of xj /xi ∈ R, say
x1 /x0 , x2 /x0 , . . . , xs /x0 . We have xs+1 /x0 ̸∈ R and hence x0 /xs+1 ∈ R. Moreover,
for any j = 1, 2, . . . , s, we have
xj /x0 ∈ R, x0 /xs+1 ∈ R ⇒ xj /xs+1 inR.
Then, we can suppose that A0 ⊂ R and then the irreducible variety Y given by the
prime ideal
A0 ∩ MR
gives the center of the valuation.
What is important for us is that the valuation defines a center in each projective
model. The usual blow-up
π :V′ →V
of V with center a subvariety is a birational morphism, that is k(V ′ ) = k(V ).
Noting Y ′ and Y the respective center of R, we have π(Y ′ ) ⊂ Y .
The assignation of center in fact determines the valuation. We shall see this in
a more detailed way in the case of dimension two.
2.3. Hardy Fields. Let us consider the set H∞ of the germs at +∞ of C ∞ real
functions of una real variable. That is, two C ∞ -functions
f : (a, +∞) → R,
g : (b, +∞) → R
represent the same element of H∞ if there is c > a, b such that f |(c,+∞) = g|(c,+∞) .
As a set of germs of functions, we have defined f + g and fg for any f , g ∈ H∞ .
Moreover, the derivative f ′ is also defined. Note that 1/f is well defined if given a
representant f of f , there is a t0 such that f (t) ̸= 0 for t > t0 .
By definition, a field H is a Hardy field if and only if R ⊂ H ⊂ H∞ and moreover
f ∈ H ⇒ f ′ ∈ H.
(The laws in H are the same as in H∞ and the inclusion R ⊂ H is interpreted via
the constant functions).
The definition seems innocent, but it contains most of the classical aspects of
the comparison between functions going to infinity.
The first observation we would like to do is that H is naturally an ordered field.
In fact, given f ∈ H with f ̸= 0, the existence of 1/f indicates that we have one of
the following possibilities
(1) For any representant f of f , there is a t0 such that f (t) > 0 for t > t0 .
(2) For any representant f of f , there is a t0 such that f (t) < 0 for t > t0 .
In the first case we say that f > 0 and in the second one f < 0. This allows us to
compare two f and g by saying
f < g ⇔ g − f > 0.
We can interpret this facts by saying that the elements of a Hardy field are non
oscillating between them.
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FELIPE CANO
The second observation is that given f ∈ H the limit
lim f = lim f
t→∞
always exists. To see this compare with the constants. Then we have three possible
situations:
(1) lim f = 0.
(2) lim f ∈ R \ {0}.
(3) lim f ∈ {−∞, +∞}.
Now, the ring
RH = {f ∈ H; lim f ̸∈ {−∞, +∞}}
is a valuation ring, naturally associated to H. The maximal ideal is given by
MH = {f ∈ H; lim f = 0}.
The units are RH \ MH = {f ∈ H; lim f ∈ R \ {0}}. Let us denote
νH : H \ {0} → ΓH = H/ ∼
the corresponding valuation. Two elements f , g ∈ H \ {0} are equivalent if and only
if ν(g/f ) = 0, that is
g(t)
lim
= c ∈ R \ {0},
t→+∞ f (t)
where f, g are representantives of f , g. This is the classical notion of equivalence
between infinite or infinitesimal functions.
The role of the derivative concerns the levels of arquimedean subgroups of ΓH ,
that levels correspond to the possible values of the logarithmic derivatives f ′ /f .
We do not insist on this for the moment.
2.4. Non oscillating trajectories and Hardy fields. Let γ, with ω(γ) = {0},
be a transcendental non oscillating trajectory of an analytic vector field ξ defined
in a neighborhood of the origin 0 ∈ R3 . We would like to understand the dynamics
given by ξ “around”, or “along” γ.
First of all, note that the fact that γ is transcendental implies that ξ is the
unique vector field having γ as a trajectory (up to multiplying by a function). This
observation, known as “Moussu’s truc” is simple and very useful in many contexts.
In fact, if there is another vector field ξ1 having γ as a trajectory, then γ is contained
in the analytic set where ξ and ξ1 give proportional tangent vectors, the fact that
γ is transcendental implies that this set is the whole space and thus ξ and ξ1 are
proportional one to the another.
Now, let K = R(X, Y, Z) be the field of rational functions of R3 . That is, the
elements of K are quotients of polynomials
ϕ=
F (X, Y, Z)
;
G(X, Y, Z)
G(X, Y, Z) ̸= 0.
Recall that each polynomial F (X, Y, Z) is identified with the polynomial function
fF : R3 → R;
(a, b, c) 7→ F (a, b, c).
Now, consider the parameterized trajectory γ : (a, +∞) → R3 with
γ(t) = (γ1 (t), γ2 (t), γ3 (t)).
SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
11
We can define an application Ψγ : K → H∞ given by
ϕ=
F (X, Y, Z)
fF ◦ γ
F (γ1 (t), γ2 (t), γ3 (t))
7→ Ψγ (ϕ) =
=
.
G(X, Y, Z)
fG ◦ γ
G(γ1 (t), γ2 (t), γ3 (t))
Since γ is transcendental, this application is well defined, since G(γ1 (t), γ2 (t), γ3 (t))
never vanishes. The image Hγ is a field and in particular it is isomorphic to K.
Moreover, the fact that γ is an integral curve of ξ and hence
γ ′ (t) = ξ(γ(t))
assures that Hγ is a Hardy field (chain’s rule). Finally, we obtain a valuation
νγ : K → Γγ by considering the composition
K → Hγ → ΓHγ = Γγ .
What is the center of this valuation in R3 ?
Note first that the values of the variables are non negative, since
lim γi (t) = 0,
t→+∞
for i = 1, 2, 3. In fact the values are positive and hence the center is the origin.
Moreover, after one blow-up, the trajectory γ has a well defined tangent p′ ∈ S2 ,
or in P2R if we take the non oriented blowing-up that, by the same arguments as
above, is the center of the valuation in the new ambient space.
Finally, performing blowing-ups at the successive centers of the valuation we
recover the iterated tangents of γ and conversely.
2.5. Galois packages. What we mean by the dynamics “around” γ?
We have seen that a valuation, as a generalization in some sense of a parameterized curve, gives a choice of infinitely near points by successive blow-ups. Thus
it guides a “polarized” vision of the phenomena in the ambient space along the
centers. In this way, the answer should be:
The dynamics around γ corresponds to the part of the space formed
by all the trajectories having the same iterated tangents as γ.
The set Gγ of all the trajectories having the same iterated tangents as γ is called
the Galois package of γ. This name comes from classical Galois Theory, let us
justify it. An integral curve is a type of “solution” of a differential equation given
by ξ and the coefficients of ξ are real analytic functions. Thus in order to distinguish
different solutions we have “right” to use operations in the analytic category (or
in the algebraic one) as blow-ups, local coordinate changes and so on. The fact of
fixing a starting point or an initial condition outside of the origin is a forbidden
operation that should produce the effect of an expansion of the coefficient field.
This is what happens with the roots of a polynomial, we have not the right to add
a given root to the coefficient field in Galois Theory. So the trajectories that have
the same iterated tangents as γ cannot be separated one from the other by the
allowed operations and thus they are in the same package of ambiguity or Galois
package.
One can ask if there are too many different ways of looking to the dynamics in
this polarized context. The answer is not so clear and here there are many things
to be clarified. To solve this kind of questions it is interesting to do a reduction of
singularities, that is blow-up and blow-up until our vector field is relatively simple
everywhere.
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FELIPE CANO
If we blow-up following a given trajectory and we arrive to a reduced situation,
as it in fact possible as we will see later, there is still a perturbing question: how
many trajectories or, more generally, how many valuations, we have to follow to
have a good global picture of the situation?
In the classical frame of algebraic geometry, this corresponds to the compactness
of the Zariski Riemann Surface Z(K): the set of all the valuations form a compact
topological space and hence any open cover admits a finite subcover. In our case
is less clear how to use this fact, but one can hope that once we have a desingularization along a valuation, the valuations near it in this topological space are also
desingularized and thus it is enough to look only at finitely many of them.
Let us end this section by giving an idea of this compactness result. Let K be a
field of rational functions of an algebraic variety over k and denote by Z(K/k) the
set of k-valuation rings of K. Put K ∗ = K \ {0} and consider the compact product
topological space
∗
{−1, 0, 1}K = {τ : K∗ → {−1, 0, 1}}.
∗
We have an application θ : Z(K/k) → {−1, 0, 1}K given by

/R
 −1, if ϕ ∈
0, if ϕ ∈ R \ MR
θ(R)(ϕ) =

1, if ϕ ∈ MR
It is an injective application and it holds that the image is closed, hence a compact
topological space.
Let us show that the image is closed.For this, we list the necessary and sufficient
∗
conditions for τ ∈ {−1, 0, 1}K to be in the image of θ. If τ = θ(R) we have that
R = {ϕ ∈ K ∗ ; τ (ϕ) ̸= −1};
MR = {ϕ; τ (ϕ) = 1}.
Thus, the conditions we need is that Rτ = {ϕ ∈ K ∗ ; τ (ϕ) ̸= −1} ∪ {0} is a valuation ring
and Mτ = {ϕ; τ (ϕ) = 1} ∪ {0} is its maximal ideal. Let us list this conditions:
(1) Rτ is a ring: If τ (ϕ) ∈ {0, 1}, τ (ψ) ∈ {0, 1}, then τ (ϕ+ψ) ∈ {0, 1}, τ (ϕψ) ∈ {0, 1}
and τ (−ϕ) ∈ {0, 1}.
(2) Rτ is a valuation ring: If τ (ϕ) = −1, then τ (1/ϕ) ∈ {0, 1}.
(3) Mτ is the only maximal ideal of Rτ :
(a) If τ (ϕ) = 1, τ (δ) = 1, τ (ψ) ∈ {0, 1}, then τ (−ϕ) = 1, τ (ϕ + δ) = 1 and
τ (ψϕ) = 1.
(b) If τ (ϕ) = 1, then τ (1/ϕ) = −1.
(c) If τ (ϕ) = 0, then τ (1/ϕ) = 0.
The above set of conditions correspond to an intersection of closed sets in {−1, 0, 1}K
and hence the image is closed. For instance, given ϕ ∈ K ∗ the set
∗
∗
{τ ∈ {−1, 0, 1}K ; τ (ϕ) = 1, τ (1/ϕ) = −1}
is a closed set. The intersection of this sets corresponds to the statement “if τ (ϕ) = 1,
then τ (1/ϕ) = −1”. (There is a slight imprecision in this proof corresponding to the case
that ϕ + ψ = 0 for instance, the reader may supply the details).
2.6. Dancing trajectories and separated packages. The reduction of singularities in dimension three of a real analytic vector field along a non oscillating
trajectory γ has the following consequence that describes the Galois package Gγ .
SISTEMAS DINÁMICOS, VALORACIONES Y EXPLOSIONES
13
The Galois package Gγ is either “separated” or “linked”. Moreover
in the case of a linked package the iterated tangents of the trajectories follow the infinitely near points of an absolutely formal curve
Γ̂ and they fulfill a semi-analytic open set of the space.
Consider two transcendental non oscillating trajectories γ and γ1 sharing the same
iterated tangents. We say that γ and γ1 are separated if there is a sub-analytic map
(analytic after blow-ups)
f : R3 → R2
such that f (γ) ∩ f (γ1 ) = ∅. Otherwise, we say that γ and γ1 are linked, “dancing
one around the other”.
The package is called separated if every pair of trajectories is separated. It is
linked if if every pair of trajectories is linked. Note that the opposite of a separated
package is not necessarily a linked package.
We shall see later an outline of the proof of the above dichotomy.
3. Valuations in dimension two. Bamboos and desingularization of a
foliation
4. Applications in dimension three. Dichotomy separation-linking for
Galois packages. Local uniformization. Global results of
Panazzolo.
5. Codimension one foliations and vector fields along a formal
curve. Examples of F. Sanz y F. Sancho
6. Ends of leaves and local Brunella’s alternative