This post describes the construction of quasianalytic algebras of functions with simple logarithmic transseries as asymptotic expansions.

The construction is based on Ilyashenko’s class of *almost regular* functions, as introduced in his book on Dulac’s problem. This class forms a group under composition, but it is not closed under addition or multiplication; to obtain a ring, we need to allow finite iterates of the logarithm in the asymptotic expansions.

Among other things, this leads to dealing with asymptotic series that have order type larger than $\omega$, so we first need to define what we mean by “asymptotic expansion”.

Let $G$ be a multiplicative subgroup of some Hardy field of $C^\infty$-germs at $+\infty$, and let $\Gs{\RR}{G}$ denote the corresponding generalized series field. (The support of such a series is a *reverse well-ordered* subset of $G$.) Let $K$ be an $\RR$-algebra of $C^\infty$-germs at $+\infty$, and let $T:K \into \Gs{\RR}{G}$ be an $\RR$-algebra homomorphism.

For $F \in \Gs{\RR}G$ and $g \in G$, we denote by $F_g$ the truncation of $F$ above $g$.

#### Definition

We say that $(K,G,T)$ is a **quasianalytic asymptotic** (or **qaa** for short) algebra if

- $T$ is injective;
- $T(K)$ is truncation closed;
- for every $f \in K$ and every $g \in G$, we have

$$

\left| f(x) – T^{-1}((Tf)_g)(x) \right| = o(g(x))

\quad\text{as } x \to +\infty.

$$

Our aim is to construct a qaa field $(\K,\G,T)$ such that $\K$ contains Ilyashenko’s class of almost regular mappings and $\G$ is the group of monomials of the form $\log_{-1}^{\alpha_{-1}} \log_0^{\alpha_0} \cdots \log_k^{\alpha_k}$, where $k \in \{-1\} \cup \NN$, $\alpha = (\alpha_{-1}, \dots, \alpha_k) \in \RR^{2+k}$ and $\log_i$ denotes the $i$th compositional iterate of $\log$ (so that $\log_0 = x$ and $\log_{-1} = \exp$).

The construction is based on a Phragmén-Lindelöf principle: for $C>0$, we define the **standard quadratic domain**

$$

\Omega = \Omega_C := \set{z + C\sqrt{1+z}:\ z \in \CC_+},

$$

where $\CC_+$ denotes the right half-plane of $\CC$.

#### Phragmén-Lindelöf Principle

(See Theorem 1 on p. 23 of Ilyashenko’s book.)

*Let $\Omega \subseteq \CC$ be a standard quadratic domain and $f:\bar\Omega \into \CC$ be holomorphic. If $f$ is bounded and, for $n \in \NN$ and $x \in \RR$,
$$
f(x) = o\left(e^{-nx}\right) \quad\text{as}\quad x \to +\infty,
$$
then $f = 0$.*

In view of the Phragmén-Lindelöf Principle, we define $\A^0_0$ to be the set of all germs at $+\infty$ of functions $f:\RR \into \RR$ that have a bounded, holomorphic extension $\f:\Omega \into \CC$ to (the closure of) some standard quadratic domain $\Omega$ and for which there exist real numbers $0 \le \nu_0 < \nu_1 < \cdots$ and $a_0, a_1, \dots$ such that $\lim_{n \to \infty} \nu_n = +\infty$ and $$ \f(z) – \sum_{n=0}^N a_n e^{-\nu_n z} = o\left(e^{-Nz}\right) \quad\text{as}\quad |z| \to \infty \text{ in } \Omega, \quad\text{for all } N \in \NN. $$ In this situation, we set $T^0_0f:= \sum_{n=0}^\infty a_n e^{-nx} \in \TT$; by the Phragmén-Lindelöf Principle, the triple $\left(\A^0_0, \G,T^0_0\right)$ is a qaa algebra.

#### Remark

The algebra $\A^0_0 \circ (-\log)$ is the algebra $\A_1$ considered in this paper.

Next, we let $\F^0_0$ be the fraction field of $\A^0_0$ and extend $T_0$ to $\F^0_0$ in the obvious way. Note that the functions in $\F^0_0$ do not all have *bounded* holomorphic extensions to standard quadratic domains; hence the need for first defining $\A^0_0$.

We now construct qaa fields $\left(\F^0_k, \G, T^0_k\right)$, for $k \in \NN$, such that $\F^0_k$ is a subfield of $\F^0_{k+1}$ and $T^0_{k+1}$ extends $T^0_k$, as follows: assuming $\left(\F^0_k,\G, T^0_k\right)$ has been constructed, we set

$$

\F^{-1}_{k+1} := \F^0_k \circ \log

$$

and define $T^{-1}_{k+1}:\F^{-1}_{k+1} \into \TT$ by $$T^{-1}_{k+1}(f \circ \log) := \left(T^0_k f\right) \circ \log.$$

Note, in particular, that every $f \in \F^{-1}_{k+1}$ has a holomorphic extension $\f:\Omega \into \CC$ on some standard quadratic domain $\Omega$ depending on $f$.

Then $\left(\F^{-1}_{k+1},\G,T^{-1}_{k+1}\right)$ is a qaa field, and we let $\A^0_{k+1}$ be the set of all germs at $+\infty$ of functions $f:\RR \into \RR$ that have a bounded, holomorphic extension $\f:\Omega \into \CC$ to some standard quadratic domain $\Omega$ and for which there exist real numbers $0 \le \nu_0 < \nu_1 < \cdots$ and germs $a_0, a_1, \dots$ in $\F^{-1}_{k+1}$ such that $\lim_{n \to \infty} \nu_n = +\infty$ and $$ \f(z) – \sum_{n=0}^N \a_n(z) e^{-\nu_n z} = o\left(e^{-\nu_Nz}\right) \quad\text{as}\quad |z| \to \infty \text{ in } \Omega, \quad\text{for all } N \in \NN. $$ In this situation, we set $T^0_{k+1} f:= \sum_{n=0}^\infty \left(T^{-1}_{k+1} a_n\right) \cdot e^{-nx} \in \TT$; by the Phragmén-Lindelöf Principle, the triple $\left(\A^0_{k+1}, \G,T^0_{k+1}\right)$ is again a qaa algebra. Finally, we let $\F^0_{k+1}$ be the fraction field of $\A^0_{k+1}$ and extend $T^0_{k+1}$ correspondingly.

#### Remarks

- $\A^0_1 \circ (-\log)$ contains all correspondance maps near hyperbolic singularities of planar real analytic vector fields, see Theorem 3 on p. 24 of Ilyashenko’s book.
- One shows, by induction on $k$, that both $\F^0_k$ and $\F^{-1}_{k+1}$ are subalgebras of $\F^0_{k+1}$, and that the restrictions of $T^0_{k+1}$ to $\F^0_k$ and $\F^{-1}_{k+1}$ are $T^0_k$ and $T^{-1}_{k+1}$, respectively.

In view of Remark 2 above, we set $\F:= \bigcup_k \F^0_k$ and $T:= \bigcup_k T^0_k$. It follows that $(\F,\G,T)$ is a qaa field and, by construction, we have $\F \circ \log \subseteq \F$.

Indeed, the algebra $\F \circ (-\log)$ of germs at $0^+$ is closed under composition, as the following implies:

#### Proposition

*Let $f,g \in \F$ be such that $g(+\infty) = +\infty$. Then $f \circ \log \circ g \in \F$.*