Let $\xi:\RR^2 \into \RR^2$ be a vector field of class $C^1$, and assume that the expansion $\RR_\xi$ of the real field by $\xi$ is o-minimal. In the the previous post, I found a $C^1$-cell decomposition $\C$, definable in $\RR_\xi$, and I introduced the forward progression map $f_\xi:B \into B$ associated to this cell decomposition.
As before, let $\C_{\textrm{trans}}$ be the set of all cells in $\C$ that are transverse to $\xi$; then $B = \bigcup\Ctrans \cup \{\infty\}$. Recall that to every one-dimensional $C \in \Ctrans$ we associate a linear ordering $<_C$ (determined by the vector field $\xi^\perp$, say, or the fact that $C$ is the graph of a $C^1$ function defined on an interval); the $<_C$-topology on $C$ then is the same as the topology induced on $C$ from that of the plane $\RR^2$.
In addition to the o-minimality of the structure $\bo$, we make the following observations. Set $S:= \left(f_\xi\right)^{-1}(\infty)$, and set $B’:= B \setminus S$ and $B”:= f_\xi(B’)$.
Facts (Dolich – S, Section 4)
- The restriction $\left(f_\xi\right)\rest{B’}:B’ \into B”$ is a bijection with inverse $b_\xi:B” \into B’$.
- Given $C,C’ \in \Ctrans$, the set $\left(f_\xi\right)^{-1}(C’)$ is a $<_C$-subinterval $I$ of $C$, the image $I’:= f_\xi(I)$ is a $<_{C’}$-subinterval of $C’$, and the restriction $\left(f_\xi\right)\rest{I}:I \into I’$ is an order-preserving or order-reversing bijection.
We extend the map $b_\xi$ obtained in Fact 1 above to all of $B$ by setting $b_\xi(z):= \infty$ for all $z \in B \setminus B”$, and we call this $b_\xi$ the backward progression map of $\xi$ associated to $\C$.
Also, given $C,C’ \in \Ctrans$, we denote the restriction of $f_\xi$ to $I$ obtained in Fact 2 by $f_{C,C’}$ and call it the correspondance map from $C$ to $C’$. To complete the picture, we also define $f_{C,\infty}$ the restriction of $f_\xi$ to $C \cap S$, for each $C \in \Ctrans$.
Remark
Each of these correspondance maps is definable in $\bo$, and the graph of $f_\xi$ is the union of the graphs of all of these correspondance maps. Hence, in practice, it suffices to study the nature of these correspondance maps when attempting to prove o-minimality of $\bo$ (i.e., Dulac’s Problem for $\xi$).
In general, not much else than what I have described in this and the previous posts is known about the correspondance maps $f_{C,C’}$. In the case where $\xi$ has a real analytic extension to the sphere $S^2$, however, they play a central role in both known proofs of Dulac’s Problem for $\xi$. In the rest of this course, I will discuss their nature in more detail using tools from real and complex analysis, the o-minimal structure $\Ranexp$ and transseries.
Therefore, we assume from now on that $\xi$ has a real analytic extension to $S^2 = \RR^2 \cup \{P\}$. (This is what I call “back to reality”.)
In this situation, we add $\{P\}$ to $\C$ and extend $f_\xi$ accordingly (depending on whether $P$ is a singular point of $\xi$ or not).
From that the flow of a real analytic vector field depends real analytically on initial conditions, we get the following:
Fact 3
Assume that $\xi$ has a real analytic extension to $S^2$. Let $C,C’ \in \Ctrans$ and $I$ and $I’$ be as in Fact 2 above, and let $J \subseteq I$ be a compact subinterval. Then the restriction of $f_{C,C’}$ to $J$ is real analytic and hence definable in $\,\Ran$.
Therefore, we will focus our attention on the germs of $f_{C,C’}$ at the endpoints of $I$. To do so, we introduce real analytic coordinates $x:[0,\epsilon) \into \cl(I)$ and $y:[0,\delta) \into \cl(I’)$, where we consider $I$ and $I’$ again as subsets of $S^2$, such that $x(0) \in \fr(I)$, $y(0) \in \fr(I’)$ and $$f_{C,C’}(x(t)) = y(t), \quad\text{for } t \in (0,\epsilon).$$ Then studying the germ of $f_{C,C’}$ near the endpoint $x(0) \in I$ comes down to studying the germ at $0^+$ of the real function $g_{C,C’}:(0,\epsilon) \into (0,\delta)$ defined by $$g_{C,C’}(t):= y^{-1}(f_{C,C’}(x(t))).$$ This function is real analytic and extends continuously to $0$, by definition, but as we shall see, it is not real analytic at $0$ in general; in particular, it is not definable in $\Ran$ in general.
From now on, I will generally refer to the functions $g_{C,C’}$ as correspondance maps of $\xi$.
In view of Theorem 7, it makes sense to state the following:
Conjecture
There is an o-minimal expansion of the real field in which all correspondance maps of real analytic vector fields on $S^2$ are definable.
This conjecture is open, and it inspires the material covered in this course.