---
abstract: |
    Following ideas of Arnold and Seigal-Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi. et al.

    In the second part of the paper we use similar ideas to prove a result on embedding formal diffeomorphisms in one-parameter groups extending a result of Takens. In particular this implies that the results of Arnold and Seigal-Yakovenko are formal consequence of our result for Lie groups.
address: 'University of Toronto, Toronto, Canada'
author:
- Gal Binyamini
bibliography:
- 'refs.bib'
title: Finiteness properties of formal Lie group actions
---

[^1]

Introduction {#sec:intro}
============

Let $M$ be a differentiable manifold and $F:M\to M$ a smooth diffeomorphism. The fixed points of $F$ play an important role in determining its dynamical properties. More generally, a point $p\in M$ is said to be *periodic* of period $k\in\Z$ if $F^k(p)=p$. The asymptotic properties of the set $k$-periodic points are one of the principal dynamical invariants of $F$.

Recall that the index of an isolated fixed point $p$ of $F$ is defined to be the number of fixed points born from $p$ after a generic small perturbation of $F$ (counted with signs according to orientation). The number of $k$-periodic points counted with the corresponding indices is a more natural topological invariant. In particular, it is homotopy invariant and can be computed at the level of homology, for instance using the Lefschetz fixed point formula.

Suppose a map $F$ is given such that the number of $k$-periodic points, counted with indices, tends to infinity. In [@ShubSull], Shub and Sullivan show that in this case the number of periodic points of the map must be infinite as well. More specifically, they prove the following.

Suppose that $F:(\R^n,0)\to(\R^n,0)$ is a $C^1$ map and $0$ is an isolated fixed point of $F^k$ for every $k\in\N$. Then the index of $0$ as a fixed point of $F^k$ is bounded as a function of $k$.

Shub and Sullivan then apply this result to give a homological criterion for the infinitude of the set of periodic points based on the Lefschetz fixed point formula.

The index of an isolated fixed point may be viewed as a topological intersection index. Indeed, if $\Delta\subset M\times M$ denotes the diagonal manifold then the index of a fixed point $p$ is equal to the intersection index of $\Delta$ and $(\id\times F)(\Delta)$ at $p$. It is natural to ask whether an analog of Shub and Sullivan’s main proposition can be given for general intersection indices.

An affirmative answer in the holomorphic setting was provided by Arnold in [@Arnold]. Consider now $F:(\C^n,0)\to(\C^n,0)$ to be a germ of a biholomorphism and let $V,W\subset(\C^n,0)$ be the germs of two analytic submanifolds of complementary dimensions. Then $F^k(V)$ is again a germ of an analytic submanifold, and assuming that it has an isolated intersection with $W$ at the origin, a local intersection multiplicity $(F^k(V),W)$ is defined.

Suppose that every element of the sequence $\mu_k:=(F^k(V),W)$ for $k\in\N$ is finite. Then the sequence is uniformly bounded.

Arnold’s proof relied on the rather nontrivial Skolem-Mahler-Lech theorem from additive number theory. In [@SeigalYakov], Seigal and Yakovenko showed that Arnold’s result could be proved using simpler and much more general Noetherianity arguments. Using this approach, they were able to generalize the result to the case of groups of formal diffeomorphisms generated by finitely many commuting diffeomorphisms and vector fields (see  for the definition of the formal diffeomorphism group $\Diff[[\C^n,0]]$ and its action on formal schemes).

More specifically, a commutative group $G\subset\Diff[[\C^n,0]]$ is said to be finitely generated if there exist finitely many formal diffeomorphisms $F_1,\ldots,F_p$ and finitely many formal vector fields $V_1,\ldots,V_q$ such that all of these generators commute in the appropriate sense, and every element of $G$ can be written as a finite product of maps $F_i^{\pm1}$ and $e^{tV_j}$ where $t\in\C$. Alternatively, $G$ is a product of a (discrete) finitely generated abelian subgroup of $\Diff[[\C^n,0]]$ and a connected (finite-dimensional) abelian Lie subgroup of $\Diff[[\C^n,0]]$ such that the two subgroups commute.

Let $G\subset\Diff[[\C^n,0]]$ be a finitely generated commutative group. Let $V,W$ be two germs of formal schemes, and define $$\mu_g : G\to\N, \qquad \mu_g = (g^*V,W).$$ Then there exists a constant $N\in\N$ depending on $V,W$ and $G$ such that $$\forall g\in G: \quad \mu_g<\infty \implies \mu_g<N.$$

Seigal and Yakovenko prove this theorem essentially by showing that the set $G_j=\{g:\mu_g>j\}$ is given by the zero locus of an ideal $I_j\subset R$, where $R$ is a certain Noetherian algebra of continuous functions on $G$. The theorem (and its various generalizations given in [@SeigalYakov]) follow by a simple Noetherianity argument.

In this paper we generalize the results of [@SeigalYakov] from the commutative case to the action of an arbitrary Lie group with finitely many connected components. Moreover, we prove a formal embedding result showing that arbitrary formal diffeomorphisms may be embedded into the action of such groups, thus showing that the results of [@Arnold] and [@SeigalYakov] follow formally from our result. A synopsis of our results and some applications and corollaries is presented in . Proofs of the key results and some further analysis is given in .

Acknowledgements
----------------

I would like to express my gratitude to Sergei Yakovenko for introducing me to this problem. I would also like to thank Askold Khovanskii and Pierre Milman for helpful discussions. Finally I wish to thank the anonymous referees for numerous helpful comments, and particularly for a suggested improvement to the presentation of the proof of Theorem \[thm:comm-gp-embedding\].

Statement of our results {#sec:statement}
========================

Notation {#sec:notation}
--------

The Lie groups considered in this paper are real Lie groups of finite dimension, possibly admitting infinitely many connected components (unless otherwise stated). Algebraic groups are always considered over $\C$.

Let $\C[[x]]$ denote the ring of formal power series in $n$ variables. Denote by $\fm\subset\C[[x]]$ the maximal ideal and by $\C_p[[x]]:=\C[[x]]/\fm^{p+1}$ the ring of $p$-jets of formal power series.

Let $\Diff[[\C^n,0]]$ denote the group of formal diffeomorphisms, i.e. $\C$-algebra automorphisms of $\C[[x]]$ which map $\fm$ to itself. Let $\Der[[\C^n,0]]$ denote the Lie algebra of formal singular vector fields, i.e. $\C$-linear derivations of $\C[[x]]$ which map $\fm$ to itself. Since $\Diff[[\C^n,0]]$ (resp. $\Der[[\C^n,0]]$) maps $\fm$ to itself, there is an induced group of $p$-jets $\Diff_p[[\C^n,0]]$ (resp. Lie algebra $\Der_p[[\C^n,0]]$), and $\Diff[[\C^n,0]]$ (resp. $\Der[[\C^n,0]]$) is the inverse limit of these groups (resp. Lie algebras). When we talk about morphisms of Lie groups, algebraic groups and Lie algebras into $\Diff[[\C^n,0]]$ or $\Der[[\C^n,0]]$ we mean that the induced maps into the $p$-jet spaces are morphisms of the prescribed type, for any $p\in\N$.

Recall that the formal exponentiation operator defines a map $$\exp:\Der[[\C^n,0]]\to\Diff[[\C^n,0]], \quad \exp(V)=e^V=I+V+\frac{V^2}{2}+\ldots$$ For any $V\in\Der[[\C^n,0]]$ there is an associated one-parameter group of formal diffeomorphisms $\tau_V:\R\to\Diff[[\C^n,0]]$ given by $\tau_V(t)=e^{tV}$. Conversely, any analytic (or even continuous) one parameter group $\tau:\R\to\Diff[[\C^n,0]]$ admits an infinitesimal generator $V\in\Der[[\C^n,0]]$, i.e. $\tau=\tau_V$ (see [@iy:book Equation 3.6]).

For $\alpha\in\N^n$, let $x^\alpha\in\C[x]$ denote the corresponding monomial in the standard multiindex notation. Let $(\cdot,\cdot)$ denote the standard inner product with respect to the monomial basis, i.e. $(x^\alpha,x^\beta)=\delta_{\alpha,\beta}$.

Noetherianity of Lie group actions
----------------------------------

Let $G$ denote a Lie group, and $\rho:G\to\Diff[[\C^n,0]]$ a homomorphism of Lie groups. We say that $G$ acts formally on $(\C^n,0)$. For $p\in\N$ we denote by $\rho_p:G\to\Diff_p[[\C^n,0]]$ the induced map.

Let $G=\R$ and $V\in\Der[[\C^n,0]]$. Then map $\tau_V:\R\to\Diff[[\C^n,0]]$ defined in  determines a formal action of $\R$ on $(\C^n,0)$. If $V$ is holomorphic in some disc containing the origin, then for $t\in\R$ the action of $t$ is given by $e^{tV}$, the holomorphic time-$t$ flow of $V$ in the usual analytic sense.

For any pair $\alpha,\beta\in\N^n$ we define the *matrix coefficient* $$\rho_{\alpha,\beta}:G\to\C, \quad \rho_{\alpha,\beta}(g) = (\rho(g)(x^\alpha),x^\beta).$$ We define *the space of matrix coefficients of $\rho$* to be the $\C$-vector space spanned by the set of all matrix coefficients of $\rho$. This space agrees with the union of the corresponding spaces of matrix coefficients of $\rho_p$ for $p\in\N$. Since the space of matrix coefficients of a finite-dimensional group representation is independent of the choice of basis elements, it follows that the space of matrix coefficients of $\rho$ is invariant under a formal change of coordinates (since it induces a linear isomorphism at the level of jets).

We are now ready to state our main result. Denote by $C(G)$ the $\C$-algebra of continuous functions on $G$.

\[thm:main\] Let $G$ be a Lie group with finitely many connected components and $\rho:G\to\Diff[[\C^n,0]]$ a Lie group homomorphism. Then the space of matrix coefficients of $\rho$ is contained in a Noetherian $\C$-algebra $R_\rho\subset C(G)$. Moreover, if $K$ is a maximal compact subgroup of $G$ then $$\dim R_\rho{\leqslant}\dim G+n(\dim G-\dim K)$$ where $\dim R_\rho$ denotes the Krull dimension of $R_\rho$.

Theorem \[thm:main\] generalizes the Noetherianity result of [@SeigalYakov] to the case of arbitrary Lie groups (with finitely many connected components) acting formally on $(\C^n,0)$. The proof of the theorem is presented in .

Local intersection dynamics
---------------------------

By a *germ of a formal scheme* we shall mean a closed subset $V\subset\spec \C[[x]]$. In other words, $V$ will be identified with an ideal $I_V$ of the ring $\C[[x]]$. Two germs of formal schemes $V,W$ of complementary dimension are said to intersect *properly* if $I_V+I_W$ is an ideal of finite codimension in $\C[[x]]$. In this case, we define their intersection multiplicity at the origin to be $$(V,W) := \dim_\C \C[[x]]/(I_V+I_W)$$

Note that this definition of multiplicity gives the correct notion when $V,W$ are complete intersections. In the general case a more delicate definition is needed to provide the proper notion of intersection multiplicities, cf. [@fulton:it Section 8]. However, for our purposes this naive definition will suffice, since it provides an upper bound for the true intersection multiplicity [@fulton:it Proposition 8.2, (a)].

Since the ring $\C[[x]]$ is Noetherian, its ideals are finitely generated. The following lemma is standard. We provide a proof for the convenience of the reader.

\[lem:mult-cond\] For any ideal $I\subset\C[[x]]$ and any $m\in\N$, the condition $\dim_\C \C[[x]]/I>m$ is equivalent to a finite number of algebraic conditions imposed on the $m$-jets of the generators of $I$.

Denote by $j_m:\C[[x]]\to \C_m[[x]]$ the natural projection. We claim first that $\dim_\C \C[[x]]/I>m$ if and only if $\dim_\C \C_m[[x]]/j_m(I)>m$.

If $\dim_\C \C_m[[x]]/j_m(I)>m$ then clearly $\dim_\C \C[[x]]/I>m$ as well. Conversely, suppose $\dim \C_m[[x]]/j_m(I){\leqslant}m$. Then $j_m(\fm^m)\subset j_m(I)$. Indeed, let $y=x_{i_1}\cdots x_{i_m}\in\fm^m$ where $1{\leqslant}i_1,\ldots,i_m{\leqslant}n$. Write $y_j=x_{i_1}\cdots x_{i_j}$ for $j=0,\ldots,m$. Since these are $m+1$ elements, they must be linearly dependent over $\C$ in $\C_m[[x]]/j_m(I)$, $$c_0 y_0 + \cdots + c_m y_m \in j_m(I), \qquad c_0,\ldots,c_m\in\C.$$ If $c_k$ is the first non-zero coefficient we have $$y_k(c_k+c_{k+1}x_{i_{k+1}}+\cdots+c_m x_{i_{k+1}}\cdots x_{i_m}) \in j_m(I).$$ The element in the parenthesis on the left hand side has a non-zero constant term, hence it is invertible in $\C_m[[x]]$, and we conclude that $y_k\in j_m(I)$ and certainly also $y\in j_m(I)$. Since this is true for any monomial $y\in\fm^m$ we have $j_m(\fm^m)\subset j_m(I)$ as claimed. It now follows by an $\fm$-completeness argument that $\fm^m\subset I$ as well [@Eis:book Proposition 7.12]. Then $\dim \C[[x]]/I=\dim \C_m[[x]]/j_m(I){\leqslant}m$ as claimed.

Let $f_1,\ldots,f_g\in I$ denote a set of generators for $I$ and $$T:\C_m[[x]]^{\oplus g} \to \C_m[[x]], \qquad T(h_1,\ldots,h_g) = h_1f_1+\ldots+h_g f_g.$$ Then $\Im T=j_m(I)$, hence the condition $\dim_\C \C_m[[x]]/j_m(I)>m$ is equivalent to the condition that the corank of $T$ is larger than $m$. Finally, one may of course express this condition by the vanishing of the appropriate minors of $T$, which may be viewed as algebraic equations involving the coefficients of $j_m(f_1),\ldots,j_m(f_g)$.

Let $G$ be a Lie group and $\rho:G\to\Diff[[\C^n,0]]$ a Lie group homomorphism. Then $G$ acts on germs of formal schemes: $g^*V$ is defined to be the germ associated to the pull-back of the ideal $I_V$ by $\rho(g^{-1})$.

We can now state and prove our result on dynamics of intersections for formal Lie group actions. The following is a relatively direct consequence of Theorem \[thm:main\].

\[thm:group-intersection\] Let $G$ be a Lie group with finitely many connected components and $\rho:G\to\Diff[[\C^n,0]]$ a Lie group homomorphism. Let $V,W$ be two germs of formal schemes, and define $$\mu_g : G\to\N, \qquad \mu_g = (g^*V,W).$$ Then there exists a constant $N\in\N$ depending on $V,W$ and $G$ such that $$\forall g\in G: \quad \mu_g<\infty \implies \mu_g<N.$$

Let $v_1,\ldots,v_s$ and $w_1,\ldots,w_t$ denote generators for the ideals $I_V$ and $I_W$ respectively. Let $R_\rho$ denote the algebra of Theorem \[thm:main\]. Then the functions mapping $g$ to the Taylor coefficients of $g^*v_i$ and $g^*w_i$ are matrix coefficients of $\rho$, hence elements of $R_\rho$. Then, by Lemma \[lem:mult-cond\] the set $\{g:(g^*V,W)>k\}$ is defined by the common zeros of an ideal $I_k\subset R_\rho$. Moreover, one can assume that $I_k\subset I_{k+1}$ (otherwise simply add to each ideal $I_k$ the union of all $I_l$ for $l<k$).

Since the algebra $R_\rho$ is Noetherian, the chain $I_k$ stabilizes at some finite index $k=N-1$. Then the set of common zeros of the ideals also stabilizes, i.e. for any $g\in G$ with finite $\mu_g$ we have in fact $\mu_g<N$.

Theorem \[thm:group-intersection\] asserts the uniform boundedness of the intersection multiplicity between a formal scheme $W$ and a certain family of formal schemes $g^*V$, where $g$ varies over a Lie group with finitely many connected components (possibly *non-compact*). We give two examples to illustrate that such a result extends neither to arbitrary analytic families (even one-dimensional), nor to arbitrary groups of diffeomorphisms (even finitely generated).

We consider $\C^2$ with the coordinates $x,y$. Let $W:=\{y=0\}$ and $V_t:=\{y=P_t(x)\}$ for $t\in\R$, where $$P_t(x) = \sum_{j=1}^\infty (e^{\frac{\pi i t}j}-e^{-\frac{\pi i t}j}) x^j.$$ Then $(V_t,W)$ is equal to the order of zero of $P_t$ at $x=0$. In particular, for $p\in\N$ prime it is easy to verify that $(V_{(p-1)!},W)=p$, and the multiplicities are therefore not uniformly bounded over $t\in\R$.

The key feature in this example is that the set of exponents $e^{\l_j t}$ appearing in the expansion of $P_t(x)$ do not form a finitely-generated semigroup under multiplication. We shall see that this does not occur in Lie group actions.

\[ex:non-solvable\] Let $g_1,g_2\in\Diff[[\C^1,0]]$ be two germs of formal diffeomorphisms with the expansions $$g_j(x) := x+c_jx^{\nu_j+1}+\cdots, \qquad c_j\neq0,\, \nu_j\in\N,\, j=1,2.$$ If $\nu_1\neq\nu_2$ then a simple computation [@iy:book Proposition 6.11] shows that the commutator $g_3:=[g_1,g_2]$ admits a similar expansion with $\nu_3=\nu_1+\nu_2$. Repeating this consideration, we see that the group $G\subset\Diff[[\C^1,0]]$ generated by $g_1,g_2$ contains diffeomorphisms with a fixed point of arbitrarily high index. Interpreting the index of a fixed point as the corresponding intersection multiplicity with the diagonal as explained in , one can use this to construct an example where the conclusion of Theorem \[thm:group-intersection\] fails.

Finite jet determination for Lie group actions
----------------------------------------------

As another application of Theorem \[thm:main\], we give an immediate proof of a result of Baouendi et. al. concerning finite jet determination in [@Baouendi:jet-determination Theorem 2.10 and Proposition 5.1].

Let $G$ be a Lie group with finitely many connected components and $\rho:G\to\Diff[[\C^n,0]]$ a continuous injective homomorphism. Then there exists $p\in\N$ such that, for any $g_1,g_2\in G$, $$j^p\rho(g_1) = j^p\rho(g_2) \quad \text{if and only if} \quad g_1=g_2$$

We note that the statement of [@Baouendi:jet-determination Proposition 5.1] refers to analytic rather than formal diffeomorphisms, and is implied by the statement above. We also note that the original formulation works in the real (rather than complex) category, but the proof is not affected by this difference.

By considering $g=g_1g_2^{-1}$, it is clearly enough to prove the existence of $p\in\N$ such that for any $g\in G$, $$j^p\rho(g) = \id \quad \text{if and only if} \quad g=e.$$ Let $R_\rho$ denote the algebra of Theorem \[thm:main\]. Then the Taylor coefficients of $\rho(g)$ are elements of $R_\rho$, and in particular the condition $j^k\rho(g)=\id$ can be expressed as the common zero locus of an ideal $I_k\subset R_\rho$, where the chain $I_k$ is increasing.

Since the algebra $R_\rho$ is Noetherian, the chain $I_k$ stabilizes at some finite index $k=p$. Then the set of common zeros of the ideals also stabilizes, i.e. the condition $j^p\rho(g)=\id$ is equivalent to the condition $\rho(g)=\id$. Since $\rho$ is an injection, the claim is proved.

Embedding in abelian Lie groups and formal flows
------------------------------------------------

In this subsection we state a theorem demonstrating that the results of Arnold [@Arnold Theorem 1] and Seigal-Yakovenko [@SeigalYakov Theorem 1] are formal consequences of Theorem \[thm:group-intersection\]. We note, to be clear, that in the cases considered in these papers our proof essentially follows the local computation of Arnold and the Noetherianity argument of Seigal-Yakovenko. However, the embedding results given in this subsection appear to be new and may be of some independent interest.

\[thm:comm-gp-embedding\] Let $\rho:\Z^p\times\R^q\to\Diff[[\C^n,0]]$ be a homomorphism of Lie groups. Then there exists an abelian Lie group $G$ with finitely many connected components (in fact, a linear algebraic group) and a homomorphism of Lie groups $\rho^*:G\to\Diff[[\C^n,0]]$ such that $\Im\rho\subset\Im\rho^*$.

The proof of Theorem \[thm:comm-gp-embedding\] is presented in .

We remark that [@SeigalYakov] uses a homomorphism of the *complex* Lie group $\Z^p\times\C^q$, whereas we state Theorem \[thm:comm-gp-embedding\] for real Lie groups.

We record a simple corollary about embeddings of a diffeomorphism in the flow of a formal vector field.

\[cor-F-embedding\] Let $F\in\Diff[[\C^n,0]]$ be a formal diffeomorphism. Then there exists some $k\in\N^+$ such that $F^k$ can be embedded in a formal flow, i.e. there exists a formal vector field $V\in\Der[[\C^n,0]]$ such that $e^V=F^k$.

Let $\rho:\Z\to\Diff[[\C^n,0]]$ be given by $\rho(1)=F$ and let $G$ and $\rho^*$ be as provided by Theorem \[thm:comm-gp-embedding\]. Let $G_0$ denote the connected component of the identity.

By construction there exists $g\in G$ such that $\rho^*(g)=F$. Letting $k=\abs{G/G_0}$ we have $g^k\in G_0$. Since $G_0$ is an abelian connected Lie group, its exponential map is surjective. Thus there exists an analytic one-parameter group $\eta^*:\R\to G_0$ such that $\eta^*(1)=g^k$. Composing with $\rho^*$ we obtain an analytic one parameter group $\eta=\rho^*\circ\eta^*:\R\to\Diff[[\C^n,0]]$ with $\eta(1)=F^k$. Then the derivative of $\eta$ at the origin gives a formal vector field $V$ satisfying $e^V=F^k$ as claimed.

We note that Corollary \[cor-F-embedding\] is purely formal: even if one starts with a holomorphic diffeomorphism $F$, we only guarantee the existence of a formal vector field $V$ exponentiating to $F^k$. The problem of embedding a holomorphic diffeomorphism in the flow of a *holomorphic* vector field is significantly more delicate and admits a non-trivial analytic obstruction. For instance, a germ of a holomorphic diffeomorphism $F:(\C,0)\to(\C,0)$ tangent to identity is embeddable in a holomorphic flow if and only if its Ecalle-Voronin modulus vanishes [@iy:book Theorem 21.31].

The anonymous referee has mentioned that [@martinet] constructs a Jordan decomposition for a formal vector field using the usual Jordan decomposition at the level of $k$-jet. A similar construction can be used to decompose the formal diffeomorphism $F$ in Corollary \[cor-F-embedding\] as a product $F=HG$ where $H$ is semi-simple, $G$ is unipotent, and $H$ and $G$ commute. This may provide an alternative approach to the proof of Corollary \[cor-F-embedding\].

In  a more accurate description of the group $G$ is provided. In particular we show how to compute $k$ above in terms of the spectrum of the linear part of $F$, and use this to give a new proof of a result of Takens [@Takens] about embeddings of formal diffeomorphisms with a unipotent linear part.

Proofs {#sec:proofs}
======

In this section we prove the two theorems stated without proof in , namely Theorems \[thm:main\] and \[thm:comm-gp-embedding\].

Proof of Theorem \[thm:main\] {#sec:proof-main}
-----------------------------

Before presenting the proof we require two results from the theory of Lie groups. The first is a decomposition theorem due independently to Iwasawa [@iwasawa] and Malcev [@malcev].

\[thm:iwasawa\] Let $G$ be a connected Lie group, and $K$ a maximal compact subgroup. Then there exist subgroups $H_1,\ldots,H_r\subset G$ isomorphic to $\R$ such that any element $g\in G$ can be decomposed uniquely and continuously in the form $$g=h_1\cdots h_r k \qquad h_i\in H_i \qquad k\in K.$$

The following linearization theorem is due to Bochner. Since we are not aware of a reference for the formal case, we present the proof below.

\[thm:bochner\] Let $K$ be a compact Lie group and $\rho:K\to\Diff[[\C^n,0]]$ a homomorphism of Lie groups. Then after a formal change of coordinates, $\rho$ is equivalent to its linear representation (mapping $g$ to the linear part $\d\rho(g)$).

Let $\mu$ denote the normalized left-invariant Haar measure on $K$. We define a formal map $U\in\C[[x]]^n$ by averaging, $$U = \int_K (\d\rho(g))^{-1}\rho(g)\d\mu(g).$$ Taking linear parts we see that $$\d U = \int_K (\d\rho(g))^{-1}\d\rho(g) = \id,$$ so $U\in\Diff[[\C^n,0]]$. Moreover, by invariance of the Haar measure, $$\begin{split}
      U\circ\rho(h) &= \int_K (\d\rho(g))^{-1}\rho(gh)\d\mu(g) \\
      &= \int_K (\d\rho(gh^{-1}))^{-1}\rho(g)\d\mu(g) = \d\rho(h) U
    \end{split}$$ where in the middle equality we translate $g\to gh^{-1}$, using the invariance of the Haar measure. We thus see that $U$ conjugates $\rho(h)$ to $\d\rho(h)$ as claimed.

We require one more lemma. While simple, this lemma is in fact the heart of the proof. The lemma is due to Arnold [@Arnold Lemma 2] (see also [@SeigalYakov Lemma 5]).

\[lem:exponent-coefficients\] Let $\rho:\R\to\Diff[[\C^n,0]]$ be given by $\rho(t)=e^{tV}$ where $V\in\Der[[\C^n,0]]$. Denote by $\l_1,\ldots,\l_n$ the spectrum of the linear part of $V$. Denote by $t$ the coordinate on $\R$.

Then the matrix coefficients of $\rho$ are contained in the ring $\C[e^{\l_1t},\ldots,e^{\l_nt},t]$.

The matrix coefficients of $\rho$ are the union of the matrix coefficients of the jets $j^p\rho$ for all $p\in\N$. It is a classical result, following easily from the Jordan decomposition, that the matrix coefficients of the matrix exponential $e^{tL}$ of a finite dimensional operator $L$ are contained in the ring $\C[e^{\mu_it},t]$ where $\mu_i$ are the elements of the spectrum of $L$. Thus the claim will be proved if we show that the spectrum of the jet $j^p\rho$ is contained in the semigroup generated by $\l_1,\ldots\l_n$ for every $p$.

We may, after a linear change of coordinates, assume that the linear part of $V$ is a lower-triangular matrix. Put the degree-lexicographic ordering on all monomials in $\C_p[[x]]$. Then the matrix representing $j^p V$ is again lower-triangular, and the diagonal entry corresponding to the monomial $x^\alpha$ is precisely $\sum \alpha_i \lambda_i$. Thus the spectrum is contained in the semigroup generated by the $\lambda_i$, as claimed.

We are now ready to present the proof of Theorem \[thm:main\].

Recall that $K$ denotes a maximal compact subgroup of $G$. Assume first that $G$ is connected. By Theorem \[thm:iwasawa\] we have for every $g\in G$ a decomposition $$\label{eq:g-decomp}
    g=h_1\cdots h_r k \qquad h_i\in H_i \qquad k\in K$$ where we may think of $h_i,k$ as continuous functions of $g$.

By Theorem \[thm:bochner\], making a formal change of coordinates we may assume that the action of $K$ is linear (recall that a formal change of coordinates does not change the space of matrix coefficients). Denote by ${\widetilde{K}}\subset\GL(n,\C)$ the image of $K$ under $\rho$. Then ${\widetilde{K}}$ is a real algebraic subgroup by a theorem of Chevalley, and clearly $\dim{\widetilde{K}}{\leqslant}\dim K$. Let ${\widetilde{K}}_\C$ denote the complexification of ${\widetilde{K}}$, which is a complex-algebraic group of complex dimension equal to $\dim{\widetilde{K}}$. Since the action of $K$ is linear, the space of matrix coefficients of $\rho\rest K$ is contained in the finitely-generated $\C$-algebra of regular functions $R[{\widetilde{K}}_\C]$ on the algebraic group ${\widetilde{K}}_\C$.

For $i=1,\ldots,r$ the subgroup $H_i$ is isomorphic to $\R$, and its action $\rho_i:H_i\to\Diff[[\C^n,0]]$ is therefore given by $\rho_i(t)=e^{tV_i}$ for some infinitesimal generator $V_i\in\Der[[\C^n,0]]$ where $t\in\R\simeq H_i$. By Lemma \[lem:exponent-coefficients\] the matrix coefficients of $\rho_i$ belong to a ring $R_i$, which is a finitely-generated $\C$-algebra of dimension at most $n+1$.

Now, $$\rho(g) = \rho(h_1)\cdots\rho(h_r)\rho(k)$$ so the matrix coefficients of $\rho(g)$ are multilinear combinations of the matrix coefficients of $\rho(h_i),\rho(k)$. In particular they belong to the ring $$R_\rho = R_1\otimes_\C \cdots \otimes_\C R_r \otimes_\C R[{\widetilde{K}}_\C]$$ which, by the above, is a Noetherian $\C$-algebra (in fact a finitely generated $\C$-algebra) of dimension bounded by $\dim G+nr$ as claimed.

For the non-connected case, one can carry out the proof above in the same manner, noting that Theorem \[thm:iwasawa\] remains true for Lie groups with finitely many connected components (see for instance [@hochschild Chapter XV]).

Proof of Theorem \[thm:comm-gp-embedding\] and further results {#sec:embedding-proof}
--------------------------------------------------------------

In this subsection we prove Theorem \[thm:comm-gp-embedding\] and give some more detailed results about embeddings of formal diffeomorphisms in formal flows.

We begin with a slightly more detailed analog of Lemma \[lem:exponent-coefficients\] for the discrete case.

\[lem:power-coefficients\] Let $F\in\Diff[[\C^n,0]]$ and denote by $\l_1,\ldots,\l_n$ the spectrum of the linear part of $F$. Let $\tau:\Z\to\Diff[[\C^n,0]]$ be given by $\tau(1)=F$. Denote by $t$ the coordinate on $\Z$.

Then:

1.  If $j^p F$ is a semisimple operator for every $p\in\N$ then the space of matrix coefficients of $\tau$ is equal to $\C[\l_1^t,\ldots,\l_n^t]$.

2.  Otherwise, the space of matrix coefficients of $\tau$ is contained in $\C[\l_1^t,\ldots,\l_n^t,t]$ and contains a function of the form $\l^{\alpha t}t=\l_1^{\alpha_1t}\cdots\l_n^{\alpha_n t} t$ for some $\alpha\in\N^n$.

The proof, which we omit, proceeds in the same way as the proof of Lemma \[lem:exponent-coefficients\]. One merely needs to replace the computation of $e^{tL}$ for a matrix $L$ in Jordan form by the equally well-known computation of $L^t$ illustrated in the following simple example.

Let $A\in\Mat_{3\times3}(\C)$ be given by $$A= \begin{pmatrix}
      \l_1&1& \\
      &\l_1& \\
      &&\l_2
    \end{pmatrix}.$$ Then for $t\in\Z$, $$A^t = \begin{pmatrix}
      \l_1^t& t\l_1^{t-1}& \\
      &\l_1^t& \\
      &&\l_2^t
    \end{pmatrix}$$ and we see that the space of matrix coefficients is contained in $\C[\l_1^t,\l_2^t,t]$ and contains the function $\l_1^t t$.

Denote by $\TDiff[[\C^n,0]]$ the group of formal diffeomorphisms whose linear part is lower-triangular with respect to the standard coordinates. We remark that, in the same way as in the proof of Lemma \[lem:exponent-coefficients\], one can check that such diffeomorphisms are automatically lower-triangular with respect to the degree-lexicographic ordering on the standard monomial basis. We denote by $\TDiff_p[[\C^n,0]]$ the corresponding group of jets. For natural numbers $q>p$ denote by $\pi_{q,p}:\TDiff_q[[\C^n,0]]\to\TDiff_p[[\C^n,0]]$ the natural projection.

For any $q\in\N$ we denote by $R_q^T:=R[\TDiff_q[[\C^n,0]]$ the ring of coordinates of $\TDiff_q[[\C^n,0]]$. For any pair of multindices $|\alpha|,|\beta|\leq$ with $\alpha{\geqslant}\beta$ (in lexicographic order) we have a coordinate function $$X_{\alpha,\beta}:\TDiff_q[[\C^n,0]]\to\C, \qquad X_{\alpha,\beta}(H):= (Hx^\alpha,x^\beta).$$ For $j=1,\ldots,n$ we denote $L_j:=X_{j,j}$ the corresponding coordinate on the diagonal. Then $R_q^T$ is generated as a ring by $X_{\alpha,\beta}$ and $L_1^{-1},\ldots,L_n^{-1}$. Indeed, it is a closed subgroup of the group of all triangular matrices in $\C_q[[x]]$ (with respect to the standard monomial basis) whose coordinate ring is generated by $X_{\alpha,\beta}$ and the inverses of all diagonal elements $X_{\alpha,\alpha}$. It remains only to note that it suffices to localize by $L_{1\ldots n}$ because in $\TDiff_q[[\C^n,0]]$ we have $X_{\alpha,\alpha}=L^\alpha$.

\[lem:Gp-Gq\] Let $F\in\TDiff[[\C^n,0]]$. Let $p$ denote the first index such that $j^pF$ is non-semisimple, or $1$ if no such index exists. For any $k\in\N$ denote by $G^*_k\subset\TDiff_k[[\C^n,0]]$ the group generated by $j^k(F)$, and by $G_k$ its Zariski closure (i.e. the algebraic group generated by $j^k(F)$).

Then for any integer $q{\geqslant}p$ the projection $\pi_{q,p}$ restricts to an isomorphism of algebraic groups $G_q\to G_p$.

We consider the case that $j^pF$ is non-semisimple. The other case is similar but simpler. Denote by $R_p,R_q$ the coordinate rings of $G_p,G_q$ respectively. Denote by $\tau_p:\Z\to G_p$ the group homomorphism satisfying $\tau_p(1)=F$, and similarly for $\tau_q$.

Since $\pi_{q,p}G^*_q=G^*_p$ by definition, $\pi_{q,p}$ restricts to a homomorphism of algebraic groups $G_q\to G_p$. Moreover since $G^*_p$ is dense in $G_p$ this map is dominant. Thus we have an induced injection $\iota:R_p\to R_q$, and we claim that it is also a surjection. Henceforth we identify $R_p$ with its image in $R_q$. The image of $\iota$ contains $X_{\alpha,\beta}$ with $|\alpha|,|\beta|{\leqslant}p$ and $L_1^{-1},\ldots,L_n^{-1}$ by definition. We will show that it also contains $X_{\alpha,\beta}$ for $|\alpha|,|\beta|{\leqslant}q$ thus concluding the proof. Let $\alpha,\beta$ be such a pair.

Denote by $\l_1,\ldots,\l_n$ the diagonal elements of the linear part of $F$. We clearly have $$L_i(j^p F^t) = \l_i^t, \qquad \forall t\in\Z.$$ We claim that there also exists a regular function $S\in R_p$ such that $$S(j^p F^t) = t, \qquad \forall t\in\Z.$$ Indeed, by Lemma \[lem:power-coefficients\], $\l^{\alpha t}t$ appears in the space of matrix coefficients of $\tau_p$. It is therefore given by a linear combination of matrix coefficients. That is, there exists a regular function $S^*\in R_p$ such that $S^*(j^pF^t)=\l^{\alpha t}t$. We may thus define $S:=S^*/L^\alpha$.

Recall that by Lemma \[lem:power-coefficients\] the space of matrix coefficients of $\tau_q$ is contained in $\C[\l_1^t,\ldots,\l_n^t,t]$. Thus there exist polynomials $P_{\alpha,\beta}$ satisfying $$X_{\alpha,\beta}(j_q F^t) = P_{\alpha,\beta}(\l_1^t,\ldots,\l_n^t,t) = (P_{\alpha,\beta}(L_1,\ldots,L_n,S))(j_q F^t)$$ In other words, $X_{\alpha,\beta}$ agrees with $P_{\alpha,\beta}(L_1,\ldots,L_n,S)$ on $G^*_q$, and hence also on $G_q$, i.e. as elements of $R_q$. Thus $X_{\alpha,\beta}$ is indeed in the image of $\iota$ as claimed.

The conclusion of Lemma \[lem:Gp-Gq\] fails if one replaces the cyclic group generated by $F$ by a group $G$ generated by two arbitrary diffeomorphisms. For instance, in Example \[ex:non-solvable\] we presented a group $G$ with two generators containing diffeomorphisms with fixed points of arbitrarily high index. Since a diffeomorphism with a fixed point of index $q>1$ corresponds to a non-trivial element in the kernel of $\pi_{q,q-1}$ we see that the conclusion of Lemma \[lem:Gp-Gq\] cannot hold for any $p\in\N$.

On the other hand, Lemma \[lem:Gp-Gq\] may still hold (for an appropriate choice of $p\in\N$) under more restrictive assumptions, for instance for commutative finitely-generated groups. One may attempt to generalize the proof by obtaining an analogous description of the rings generated by the matrix coefficients of such groups. We do not pursue this further in this paper, and use an alternative approach to prove the embedding result for groups with several generators.

We now state the key proposition on the embedding of a formal diffeomorphism in an abelian linear algebraic group.

\[prop:diff-embed\] Let $F\in\Diff[[\C^n,0]]$. Then there exists an abelian linear algebraic group $G$ and a homomorphism of Lie groups $\rho:G\to\Diff[[\C^n,0]]$ such that $F\in\Im\rho$.

Additionally, if a formal diffeomorphism $F'\in\Diff[[\C^n,0]]$ commutes with $F$ then it also commutes with any element of $\Im\rho$.

After a linear change of coordinates we may assume that $F$ belongs to $\TDiff[[\C^n,0]]$. We may thus apply Lemma \[lem:Gp-Gq\]. In the notations of the lemma, for any $q{\geqslant}p$ we denote $\rho_q:G_p\to G_q$ the isomorphism of algebraic groups inverse to $\pi_{q,p}$. If $q'{\geqslant}q{\geqslant}p$ then $\pi_{q',q}\circ\rho_{q'}=\rho_q$. Thus $\rho_q:q{\geqslant}p$ form an inverse system with a limit $\rho:G_p\to \TDiff[[\C^n,0]]$, which is a homomorphism of Lie groups by definition. Moreover, since $\rho_q(j^p(F))=j^q(F)$ for any $q{\geqslant}p$, we have $\rho(j^p(F))=F$. The claim thus follows with $G=G_p$.

Finally, suppose $F'\in\Diff[[\C^n,0]]$ commutes with $F$. Then for any $q{\geqslant}p$ the jet $j^q F'$ commutes with $j^q F$, hence with $G^*_q$, and hence also with $G_q$. Since $G_q=\Im \rho_q=j^q(\Im\rho)$ we see that $F'$ commutes with $\Im\rho$ at the level of $q$-jets for any $q{\geqslant}p$, so it in fact commutes with $\Im\rho$ as claimed.

\[rem:G-structure\] From the proof above it is easy to find the structure of the group $G$. Denote by $\C_a,\C_m$ the additive and multiplicative groups of $\C$ respectively and by $D\subset\GL(n,\C)$ the group of diagonal matrices. Let $F_s$ denote the semisimple part of the linear part of $F$. After a linear change of coordinates we may assume $F_s\in D$.

With these notations $G$ is isomorphic to either $T$ or $T\times\C_{\text{a}}$ (for the semisimple and non-semisimple cases in Lemma \[lem:Gp-Gq\], respectively) where $T\subset D$ is the algebraic group generated by $F_s$. Denote by $T_0$ the connected component of the identity in $T$. The number of connected components of $G$ is given by the size of the cyclic group $T/T_0$, which can be expressed in terms of the spectrum $\l_1,\ldots,\l_n$ of $F_s$ as follows. Let $\Phi$ denote the group of multiplicative characters $\chi:D\to\C_m$ such that $\chi(F_s)$ is a root of unity. Then $T_0$ is the common kernel of the characters in $\Phi$, and consequently $$\label{eq:T-conn-comp}
    \#(T/T_0) = \#\{\chi(F_s):\chi\in\Phi\}.$$ Finally, the right hand side of  is the size of the group of roots of unity inside the multiplicative group generated by $\l_1,\ldots,\l_n$.

Theorem \[thm:comm-gp-embedding\] is now an easy corollary.

We prove the claim by induction on $p$. Let $\rho:\Z^p\times\R^q\to\Diff[[\C^n,0]]$ be a homomorphism of Lie groups. By the inductive hypothesis, there exists an abelian linear algebraic group $G$ and a homomorphism of Lie groups $\rho^*:G\to\Diff[[\C^n,0]]$ such that $\rho(\{0\}\times\Z^{q-1}\times\R^q)\subset\Im\rho^*$, and $\rho(1,0,\ldots,0)$ commutes with $\Im\rho^*$.

By Proposition \[prop:diff-embed\] there exists an abelian linear algebraic group $G'$ and a homomorphism of Lie groups $\rho':G'\to\Diff[[\C^n,0]]$ such that $\rho(1,0,\ldots,0)\in\Im\rho'$ and $\Im\rho^*$ commutes with $\Im\rho'$. Then the group $G\times G'$ with the homomorphism $\rho^*\times\rho'$ satisfies the required conditions.

From Remark \[rem:G-structure\] and the proof of Corollary \[cor-F-embedding\] we have the following corollary. In particular, this extends a result of Takens [@Takens] stating the a formal diffeomorphism with a unipotent linear part can always be embedded in the flow of a formal vector field.

Let $F\in\Diff[[\C^n,0]]$ be a formal diffeomorphism. Let $\l_1,\ldots,\l_n$ denote the spectrum of the linear part of $F$, and let $k$ denote the size of the group of roots of unity in the multiplicative group generated by $\l_1,\ldots,\l_n$.

Then $F^k$ can be embedded in a formal flow, i.e. there exists a formal vector field $V\in\Der[[\C^n,0]]$ such that $e^V=F^k$.

[^1]: The author was supported by the Banting Postdoctoral Fellowship and the Rothschild Fellowship
