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 $f \circ \log \circ g \in \F$.