Let $\xi:S^2 = \RR^2 \cup \{P\} \into \RR^2$ be a real analytic vector field. Let $\C$ be a cell decomposition definable in $\Ran$ as obtained for $\xi\rest{\RR^2}$, and enlarge $\C$ by adding $\{P\}$.

### Exercise 1

Use o-minimality to show that all but fginitely many singularities of $\xi$ are removable.

Based on Exercise 1, we assume from now on that $\xi$ has finitely many singularities. In this situation, we hope to find an informative description of the correspondance maps $g:(0,\epsilon) \into (0,\delta)$.

Using Dulac’s (and more generally, Dumortier’s) resolution of singularities result, as treated in Panazzolo’s module for this class, we may assume that every singularity of $\xi$ is **elementary**. Letting $J(\xi)(z)$ denote that Jacobian matrix of $\xi$ at the point $z$, this means that $J(\xi)(z)$ has at least one nonzero eigenvalue, for every $z \in Z(\xi)$. In this case, Ilyashenko is able to describe the nature of each correspondance map $g$ in such a way as to conclude that Dulac’s Problem holds for $\xi$.

In what follows, we will focus out attention on a special, but generic, case:

### Definition

A singular point $z$ of $\xi$ is **hyperbolic** if the product of the two eigenvalues of $J(\xi)(z)$ is negative.

(Note that the eigenvalues of $J(\xi)(z)$ are the roots of the real characteristic polynomial of $J(\xi)(z)$, so they are are either complex conjugates, or they are both real. In the former case, their product is a positive real number; in the hyperbolic case, they are both real, nonzero, and of opposite sign.)

Hyperbolic singularities are generic for real analytic vector fields, in the following sense: given a real analytic family $\{\xi_\mu:\ \mu \in M\}$ of vector fields on $S^2$, and given a parameter $\mu_0 \in M$ such that $\xi_{\mu_0}$ has only hyperbolic singularities, there is an open neighbourhood $U$ of $\mu_0$ in $M$ such that $\xi_\mu$ has only hyperbolic singularities for all $\mu \in U$.

The nature of correspondance maps near a hyperbolic singular point of $\xi$ is described by the following:

### Fact 2 (see Ilyashenko-Yakovenko, Theorem 24.38)

*Every correspondance map near a hyperbolic singularity of $\xi$ is almost regular.*

I will not prove this fact here, but I will define next what “almost regular” means. For this, we need to talk first about certain domains in the complex plane.

### Definition

A subset $\Omega$ of $\CC$ is a **standard quadratic domain** if $$\Omega = \Omega_C:= \varphi_C(\CC^+),$$ where $\CC^+$ is the open right halfplane, $C>0$ and $\varphi_C:\CC^+ \into \CC$ is the holomorphic map $$\varphi_C(z):= z + C\sqrt{1+z}.$$ Note that $\varphi_C:\CC^+ \into \Omega_C$ is biholomorphic.

Next, we need to introduce certain transseries called Dulac series.

### Definition

A **Dulac series** is a transseries of the form $$c_0 e^{-\alpha_0 z} + \sum_{n=1}^\infty p_n(z) e^{-\alpha_n z},$$ where

- $c_0>0$;
- $0 < \alpha_0 < \alpha_1 < \cdots$ and $\lim_{n \to \infty} \alpha_n = \infty$; and
- $p_n(z) \in \RR[z]$ for $n=1,2,\dots$.

We think of Dulac series as “series at +\infty”, in the sense that we intend use them as asymptotic expansions for germs of functions at $+\infty$.

### Exercise 2

Show that the set of all Dulac series is **closed under log-composition**, that is, if $F$ and $G$ are Dulac series, then so is the transseries $F \circ (-\log) \circ G$.

### Definition

Let $f:(0,\epsilon) \into (0,\delta)$ be a real analytic function such that $\lim_{t \to 0^+} f(t) = 0$. We call $f$ **almost regular** if there exist

- a Dulac series $T(f) = c_0 e^{-\alpha_0 z} + \sum_{n=1}^\infty p_n(z) e^{-\alpha_n z}$,
- a standard quadratic domain $\Omega$ and a holomorphic map $\bar f:\Omega \into \CC$, called a
**holomorphic continuation of $f$ in the logarithmic chart**,

such that the following hold:

- $\bar f (x) = f(e^{-x})$ for all sufficiently large $x>0$;
- the image $-\log \circ \bar f(\Omega)$ contains a standard quadratic domain $\Delta$, and for every standard quadratic domain $\Delta’ \subseteq \Delta$, the preimage of $\Delta’$ under $-\log \circ \bar f$ contains a standard quadratic domain; and
- for every $n \in \NN$, we have $$\bar f(z) – \left( c_0 e^{-\alpha_0 z} + \sum_{i=1}^n p_i(z) e^{-\alpha_i z} \right) = o\left( e^{-\alpha_n \re(z)} \right) \quad\text{as } \re(z) \to +\infty \text{ in } \Omega.$$

Point 3 means that the Dulac series $T(f)$ is an asymptotic expansion of $\bar f$ as $\re(z) \to +\infty$ in $\Omega$; in particular, $T(f)$ is uniquely determined as a Dulac series.

For example, if $f$ is a real analytic germ at $0$ such that $f(0) = 0$, then $f$ is almost regular. However, for general almost regular germs $f$, the Dulac series $T(f)$ need not converge.

### Remark

Let $f$ and $g$ be two almost regular germs with holomorphic continuations $\bar f$ and $\bar g$ in the logarithmic chart. Then, by point 2 in the definition above, the **log-composition **$\bar f \circ (-\log) \circ \bar g$ makes sense, and it is a holomorphic continuation of $f \circ g$ in the logarithmic chart.

Combining this remark with Exercise 2, we obtain the following:

### Corollary 3 (see Ilyashenko-Yakovenko, Lemma 24.33)

*The set $\A$ of all almost regular germs at $0^+$ is closed under composition.* $\qed$

### Exercise 3

Assume that every singularity of $\xi$ is hyperbolic. Based on Fact 2 above, show that the graph of each finite compositional iterate of the forward progression map $f_\xi$ is a finite union of graphs of almost regular functions.

The key observation needed to conclude that Dulac’s Problem holds if $\xi$ only has hyperbolic singularities is the following:

### Theorem 4 (see Ilyashenko-Yakovenko, Theorem 24.29)

*The set $\A$ of all almost regular germs is quasianalytic, that is, the map $T:\A \into \TT$ is injective.*

I refer you to Section 24G of Ilyashenko and Yakovenko’s book for the proof of this theorem (which we discussed in class). Most important for the purposes of this course is the following ingredient in this proof:

### Uniqueness Principle (see Ilyashenko-Yakovenko, Lemma 24.37)

*Let $\Omega$ be a standard quadratic domain and $f:\Omega \into \CC$ be holomorphic with continuous extension to the closure $\bar\Omega$ of $\Omega$. Assume there exist $M,\mu>0$ such that*

*$|f(z)| \le M$ for $z \in \partial\Omega$;**$f(z) = O\left(e^{\mu |z|}\right)$ for $z \in \Omega$; and**$f(x) = o\left(e^{-\rho x}\right)$ as $x \to +\infty$ in $\RR$, for all $\rho > 0$.*

*Then $f = 0$.*

Finally, we have the following observation, already present in Dulac’s paper:

### Exercise 4

Let $f \in \A$, and assume that the fixed points of $f$ accumulate at $0$. Show that $T(f) = e^{-z}$.

### Corollary 5

*If $\xi$ has only hyperbolic singularities, then Dulac’s Problem holds for $\xi$.*

*Proof.* Exercise 4 and Theorem 4 imply that every almost regular germ has finitely many isolated fixed points. It follows from Exercise 3 that every finite compositional iterate of the forward progression map $f_\xi$ has finitely many isolated fixed points. $\qed$

In the spirit of this post, I would like to prove that $\A$ generates an o-minimal expansion of the real line. Indeed, since real analytic germs $f$ at $0$ such that $f(0) = 0$ belong to $\A$, we may as well claim the following:

### Conjecture

*The expansion of $\Ran$ by all germs in $\A$ is o-minimal.*

This conjecture is still open. A “baby version” of this conjecture, however, was proved in this paper, where we show that the subset of $\A$ consisting of all germs $f$ whose Dulac series $T(f)$ involves only constant polynomials $p_i$ does indeed generate an o-minimal expansion of $\Ran$.

Since the only known methods for proving o-minimality involve adding quasianalytic *algebras* of functions to the real field (see Rolin and Servi’s paper), a first step towards establishing the conjecture is to show that $\A$ is contained in such an algebra.

Note that $\A$ is not an algebra: the difference of two almost regular germs $f$ and $g$ may not have a Dulac series as asymptotic expansion, because $T(f)$ and $T(g)$ may have the same leading term.

However, if we weaken the requirement that the asymptotic expansion of an almost regular germ be a Dulac series, we get into trouble with log-composition, as it may introduce terms involving powers of finite iterates of $\log$ in the resulting transseries.

Next time, then, we will show that allowing transseries involving finite iterates of $\log$, we can indeed construct a quasianalytic algebra that contains $\A$ and is closed under log-composition.