Ilyashenko algebras: putting it all together

Let $f = (f_0, \dots, f_k)$ be such that each $f_i \in \H$ is infinitely increasing and $f_0 \gt \cdots \gt f_k$. To see what it takes to generalize our construction of the Ilyashenko algebra $(\F,L,T)$ to more general monomials $f$, recall the construction in the following schematic: $$\begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_0 \\ e^{-x} \circ (x) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ \begin{bmatrix} \F_1′ \\ e^{-x} \circ (\log) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_1 \\ e^{-x} \circ (x,\log) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ & \vdots & \\ & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_{k-1} \\ e^{-x} \circ (x,\log, \dots, \log_{k-1}) \end{bmatrix} \\ & \swarrow \circ\log \swarrow & \\ \begin{bmatrix} \F_k’ \\ e^{-x} \circ (\log, \dots, \log_{k}) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_k \\ e^{-x} \circ (x,\log, \dots, \log_{k}) \end{bmatrix} \\ \end{matrix}$$ In this schematic, each bracket $\begin{bmatrix} \F_i \\ e^{-x} \circ (x,\log, \dots, \log_{i}) \end{bmatrix}$ indicates the field $\F_i$ constructed at stage $i$ whose asymptotic expansions have support generated by the germs $e^{-x} \circ (x,\log, \dots, \log_{i})$. The $\swarrow \circ\log \swarrow$ line indicates shifting this field by composing on the right by $\log$, to obtain the field $\begin{bmatrix} \F_{i+1}’ \\ e^{-x} \circ (\log, \dots, \log_{i+1}) \end{bmatrix}$ on the left of the next lower row. Finally, the $\xrightarrow{\text{(UP)}}$ arrow indicates constructing the next field with asymptotic expansions in the monomials generated by $e^{-x}$ with coefficients in $\F_{i+1}’$, using the Uniqueness Principle, and then replacing each of the coefficients in $\F_{i+1}’$ by their previously constructed asymptotic expansions. To adapt this schematic to a general tuple $f = (f_0, \dots, f_k)$, it would have to look like this: $$\begin{matrix} \RR & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \F_0 \\ e^{-x} \circ (x) \end{bmatrix} \\ & \swarrow \circ \left(f_k \circ f_{k-1}^{-1}\right) \swarrow & \\ \begin{bmatrix} \K_1′ \\ e^{-x} \circ \left(f_k \circ f_{k-1}^{-1}\right) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_1 \\ e^{-x} \circ \left(x,f_k \circ f_{k-1}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ\left(f_{k-1} \circ f_{k-2}^{-1}\right) \swarrow & \\ & \vdots & \\ & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_{k-1} \\ e^{-x} \circ \left(x,f_2 \circ f_{1}^{-1}, \dots, f_k \circ f_{1}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ \left(f_1 \circ f_{0}^{-1} \right) \swarrow & \\ \begin{bmatrix} \K_k’ \\ e^{-x} \circ \left(f_1 \circ f_{0}^{-1}, \dots, f_k \circ f_{0}^{-1}\right) \end{bmatrix} & \xrightarrow{\text{(UP)}} & \begin{bmatrix} \K_k \\ e^{-x} \circ \left(x,f_1 \circ f_{0}^{-1}, \dots, f_k \circ f_{0}^{-1}\right) \end{bmatrix} \\ & \swarrow \circ f_0 \swarrow & \\ \begin{bmatrix} \K_{k+1}’ \\ e^{-x} \circ \left(f_0, \dots, f_k\right) \end{bmatrix} &\end{matrix}$$ Provided the construction can be carried out like this for $f$, we then set $\K_f:= \K_{k+1}’$. To carry out the construction for $f$, it has to satisfy the following two conditions (see also the discussion in this post): recall that, for a tuple $h \in \H^{>0}$, $\langle h \rangle^\times$ denotes the multiplicative $\RR$-vector subspace of $\H^{>0}$ generated by $h$. $(1)_f$ for $1 \le i < j \le k$ and every standard power domain $V$, there exist a standard power domain $U$ and a holomorphic extension $\h_{ij}$ of $f_j \circ f_i^{-1}$ on $U$ such that $\h_{ij}(U) \subseteq V$; $(2)_f$ for $0 \le i \le k$, the set $\left\langle e^{-x} \circ \left(f_i \circ f_{i}^{-1}, \dots, f_k \circ f_{i}^{-1}\right) \right\rangle^\times$ is a scale on standard power domains.

Definition

We call $f$ admissible if $(1)_f$ and $(2)_f$ hold.

Example

Since $L$ is a scale on standard power domains, and since $\llog$ maps every standard power domain into every standard power domain (as germs of domains at $\infty$), every tuple $(x,\log, \dots, \log_k)$ is admissible. Our favourite examples of admissible tuples are of the following nature: recall that the set $\M$ of all exponential monomials of the set $\F$ of germs defined by $\Lanexp$-terms is a multiplicative subgroup of $\H^{>0}$ that is closed under left-composition with $\exp$. Set $$\la:= \bigcup_{k \in \NN} \M \circ \log_k,$$ the set of all monomials of $\H$.

Corollary

$\la$ is a multiplicative subgroup of $\H^{>0}$ that is closed under left-composition with $\exp$ and right-composition with $\log$; in particular, $\la$ is a multiplicative $\RR$-vector subspace of $\H^{>0}$. $\qed$ Set $$\U:= \set{-\log \circ m: m \in \la}.$$

Remarks

1. $\U$ is the set of all $h in \H$ all of whose principal monomials are large; the germs in $\U$ are called purely infinite.
2. $\H = \U \oplus \bo$, where $\bo$ is the set of all bounded germs of $\H$; in particular, every Archimedean class of $\H^{>0}$ has a unique representative in $\la$.
The construction for $\la$ is made possible by the

If each $\,f_i \in \U$, then $\,f$ is admissible. Show sketch of proof
It follows from Remark 1 above that every germ in $\U$ is simple.
Assume $f_i \in \U$ for each $i$, and fix $1 \le i \lt j \le k$. It follows from Corollary 2 of this post that $$\eh\left(f_j \circ f_i^{-1}\right) \le \max{0, \level(f_j) – \level(f_i)} \le 0.$$

By the Extension Theorem, this means that $f_j \circ f_i^{-1}$ has a holomorphic extension $h$ on some $(-1)$-domain; however, to obtain admissibility, we need a more subtle consequence of the Extension Theorem:

Proposition

Let $h \in \I$ be such that $\eh(h) \le 0$ and $h \preceq x$. Then for every standard power domain $V$, there exist standard power domains $U_1$ and $U_2$ such that $$\h(U_1) \subseteq V \quad\text{and}\quad \h(V) \subseteq U_2.$$

Since $\eh\left(f_j \circ f_i^{-1}\right) \le 0$ and $f_j \circ f_i^{-1} \lt x$, condition $(1)_f$ follows from the Proposition.

As to condition $(2)_f$, let $m in \left\langle e^{-x} \circ \left(f_i \circ f_{i}^{-1}, \dots, f_k \circ f_{i}^{-1}\right) \right\rangle^\times$ be bounded. Since $\left\langle e^{-x} \circ \left(f_i \circ f_{i}^{-1}, \dots, f_k \circ f_{i}^{-1}\right) \right\rangle^\times$ is a multiplicative group, it suffices to show that the holomorphic extension $\m$ of $m$ is bounded on every standard power domain. By definition, $\log \circ m$ is an (additive) $\RR$-linear combination of $f_i \circ f_{i}^{-1}, \dots, f_k \circ f_{i}^{-1}$; therefore, we have $\eh(-\log \circ m) \le 0$ and $-\log \circ m \preceq x$. Moreover, by Remark 2 above, either $m = 1$ or $m$ is small. In the former case $m = 1$, which is bounded, so assume $m$ is small. Then $-\log \circ m$ is infinitely increasing, so the proposition above implies that the holomorphic extension $\n$ of $-\log \circ m$ maps standard power domains into standard power domains. Therefore, on standard power domains we have $m = \eexp \circ (-n)$, which is bounded.
$\qed$

By the Admissibility Lemma for monomials, if each $f_i$ belongs to $\U$, we obtain a qaa field $(\K_f, \la, T_f)$ that extends every qaa field $(\K_g, \la, T_g)$ with $g$ a subtuple of $f$. Thus, the subtuple ordering induces a partial ordering on the collection of all $(\K_f,\la,T_f)$, with $f \in \U^{k+1}$ large and $k \in \NN$. We let $(\K,\la,T)$ be the corresponding direct limit.

Final remarks

1. Note that $L \subseteq \la$; therefore, the qaa field $(\K,\la,T)$ extends $(\F,L,T)$.
2. It follows from our description of the germs in $\H$ that every $h \in \H$ is given by a convergent $\la$-generalized power series. Indeed, there is a field homomorphism $S:\H \into \Gs{\RR}{\la}$ such that, for $h \in \H$, the image $S(h)$ is the unique convergent $\la$-generalized power series defining $h$. It follows that $(\H,\la, S)$ is a qaa field and that $(\K,\la,T)$ extends $(\H,\la,S)$.

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