In order to address some of the questions raised in this post, I introduce here some relevant definitions and recast the construction of Ilyashenko algebras based on $\log$ monomials using these new notions.
As before, let $\H$ be the Hardy field of germs at $+\infty$ of all $h:\RR \into \RR$ definable in $\Ranexp$, and denote by $\C$ the ring of all germs at $+\infty$ of continuous functions $h:\RR \into \RR$.
Denote by $\H^{>0}$ the multiplicative $\RR$-vector space of all positive elements of $\H$, and fix an arbitrary multiplicative $\RR$-vector subspace $M$ of $\H^{>0}$. Let $\Gs{\RR}{M}$ denote the corresponding generalized series ring as in van den Dries et al.
The guiding question here is: what conditions on $M$ guarantee that series in $\Gs{\RR}{M}$ can be used as asymptotic expansions of germs in $\C$ and yield a corresponding quasianalytic algebra?
I will use the dominance relation $\preceq$, as in Aschenbrenner and van den Dries, and the corresponding Archimedean equivalence relation $\asymp$ to describe asymptotic behaviour.
To simplify many things, I first want $M$ to satisfy the following:
Definition
$M$ is a pure scale if $m \not\asymp 1$ for all $m \in M$ different from $1$.
Another way to say this is that the map $\H \into \H/_\asymp$, which maps each $h \in \H$ to its archimedean class, is injective on $M$.
Examples
- For $i \in \NN$, set $$L_i:= \langle \exp = \log_{-1}, x = \log_0, \log, \dots, \log_{i-1}\rangle^\times,$$ where $\log_k$ denotes the $k$-th compositional iterate of $\log$ and $\langle A \rangle^\times$ denotes the multiplicative $\RR$-vector space generated by $A \subseteq H^{>0}$. Then each $L_i$ is a pure scale.
- Set $$L:= \bigcup_{i \in \NN} L_i;$$ then $L$ is also a pure scale.
I assume in this post that $M$ is a pure scale.
The most obvious series in $\Gs{\RR}{M}$ that can be asymptotic expansions are those with natural support: a set $S \subseteq M$ is $M$-natural if $S \cap (m,+\infty)$ is finite for every $m \in M$.
For $F = \sum a_m m \in \Gs{\RR}{M}$ and $n \in M$, denote by $$F_n := \sum_{m \ge n} a_m m$$ the truncation of $F$ above $n$. Note that if $F \in \Gs{\RR}{M}$ has $M$-natural support, then $F_n \in \C$ for every $n \in M$.
Definition
A germ $f \in \C$ has asymptotic expansion $F \in \Gs{\RR}{M}$ if $\supp(F)$ is $M$-natural and $$f – F_n \prec n \quad\text{for every } n \in M.$$
Example
Let $f \in \C$ be such that $f \circ (-\log)$ is the germ of a transition map near a hyperbolic singularity of a real analytic vector field on the plane $\RR^2$. Then by Dulac, the germ $f$ has an asymptotic expansion in $\Gs{\RR}{L_1}$.
Lemma 1
- The set $\C(M)$ of all $f \in \C$ that have an asymptotic expansion in $\Gs{\RR}{M}$ is an $\RR$-algebra.
- Every $f \in \C(M)$ has exactly one asymptotic expansion $T(f)$ in $\Gs{\RR}{M}$, and the map $T:\C(M) \into \Gs{\RR}{M}$ is an $\RR$-algebra homomorphism.
Here is one way to extend the notion of asymptotic expansion to any series in $\Gs{\RR}{M}$: a set $S \subseteq \Gs{\RR}{M}$ is truncation closed if for every $F \in S$ and $n \in M$, the truncation $F_m$ belongs to $S$.
Definition
Let $\K \subseteq \C$ be an $\RR$-subalgebra and $T:\K \into \Gs{\RR}{M}$ be an $\RR$-algebra homomorphism. The triple $(\K,M,T)$ is a quasianalytic asymptotic algebra (or qaa algebra for short) if
- (i)
- the map $T$ is injective (quasianalyticity);
- (ii)
- the image $T(\K)$ is truncation closed;
- (iii)
- for $f \in \K$ and $n \in M$, we have $f – T^{-1}\left((Tf)_n\right) \prec n$.
Examples
Let $\G$ be an $\RR$-algebra of germs at $0^+$ of $C^\infty$ functions, let $T:\G \into \Ps{R}{X}$ be the Taylor expansion map at 0, and assume that $\G$ is quasianalytic, that is, that $T$ is injective. (Examples of such $\G$ are the ring of all real analytic germs, quasianalytic Denjoy-Carleman classes, Gevrey classes, etc.) Then the triple $\left(\G \circ \exp^{-1}, L_0, T \circ \exp^{-1}\right)$ is a qaa algebra.
On the other hand, denote by $\A$ Ilyashenko’s class of almost regular functions and, for $f \in \A$, denote by $\sigma(f)$ the Dulac series asymptotic to $f$. Then the triple $(\A,L_1,\sigma)$ is not a qaa algebra, because $\A$ is not closed under addition. However, $\A$ is closed under $\log$-composition, that is, given $f,g \in \A$ such that $\frac1g \succ 1$, the composition $f \circ (-\log) \circ g$ belongs to $\A$.
One of the motivations for the definitions in this post is the following:
Theorem
There exists a qaa field $(\F,L,T)$ such that $\A \subseteq \F$, and such that $\exp, \log \in \F$ and $\F$ is closed under differentiation and $\,\log$-composition.
The proof of this theorem is outlined in this post; in the next post, I will recast this proof in the setting of the general $M$ fixed here. In subsequent posts, I will discuss what $M$—other than the $L$ above—this proof can be adapted to.